According to Scooby-Doo Behind the Scenes (released in 1998; see Wikiquote), the initial money to start out was given to them by Daphne’s dad:

Daphne: "Well, Daddy gave us the money to start off. (shows an image of the check that first financed Mystery Inc.) We didn't even have a car, so we gave our parents gas money to drive us around."

In addition, many of the different canons have shown them having jobs. (In some of these they are adults though.) From the Wikia page for Scooby-Doo:

• Junior detectives-for-hire: Contrary to the above, when the gang was in their preadolescence, they worked out of a clubhouse, calling themselves the Scooby-Doo Detective Agency, and charged a minimal fee for their services in solving the mysteries in their hometown of Coolsville. (APNSD)
• Actor: Scooby got the role of Sandy Duncan's dog, after solving the case at Mammoth Studios. (TNSDMovies: Sandy Duncan's Jekyll and Hyde)
• Carnival owner: Scooby, along with the rest of the gang, was given a share of Dick Van Dyke's carnival, after saving it from going under. (TNSDMovies: The Haunted Carnival)
• Construction worker: He, Shaggy, and Fred took questionable jobs at a construction site. (TSDS: High Rise Hair Raiser)
• Freelance journalist: Scooby, Shaggy, Scrappy, and Daphne became freelance journalists, forming the second incarnation of the Scooby-Doo Detective Agency. (TNS&SDS, TNSDMysteries)
• Gym teacher: He, Shaggy, and Scrappy were hired as gym teachers at Grimwood's Finishing School for Girls, or so they believed. (Scooby-Doo and the Ghoul School)
• Co-racecar driver. (Scooby-Doo and the Reluctant Werewolf)
• Airport customs: He and worked in customs at an airport during the break-up of Mystery Inc., but were fired when they ate all the confiscated cheese. (Scooby-Doo On Zombie Island)
• Detective: He, Shaggy, and Scrappy worked for Shaggy's uncle Fearless Shagaford at the Fearless Detective Agency. (The Scooby & Scrappy Doo Puppy Hour)
• Additionally, Scooby assisted Shaggy when he was a paper boy, by unloading the delivered papers into his bike's basket. When the gang discovered Shaggy's boss stole his bike chain to work his counterfeiting machine, Shaggy quit, and so did Scooby. (APNSD: A Bicycle Built for Boo!)

Something else that might impact things:

Scooby Doo MAY be set in a period of economic recession.

It is never outright said in the show or by any of the creators, but there is a lot of evidence that points towards the Scooby Doo universe being in the middle of an economic recession:

(source: http://scoobyfiles.toonzone.net/monsters/index.html, as well as a number of other articles that pop up when googling "Scooby Doo recession")

1. Nearly every episode was set in some sort of abandoned place where there would otherwise be a lot of people: an airport, mansions, castles, a ski resort, an amusement park,... Every location looks broken down, unused and abandoned, like the economy just doesn't support the activities they encompass.
2. Nearly every criminal was motivated by monetary greed: a rare coin, a family treasure, a smuggling operation, theft for resale,... And even those that weren't motivated directly by monetary greed just wanted a job, or in one case were motivated by hatred against robots, who tend to steal jobs. There were no crimes of passion or crimes of hatred of other people, which means that most people just wanted to get money.
3. Most of the criminals had skills that could easily land them a job: several PhD s, lawyers, engineers, scientists, entrepreneurs,... People that could easily legitimately make money using their skills in legal manners,... If there were any jobs around.
4. They don't seem to make all that much money for food and other necessities. In a number of episodes, Scooby and Shaggy react very hungrily when they encounter large amounts of food, as if they don't know when they next get food. And they seem to easily be bribed by relatively small amounts of Scooby Snacks from time to time. And look at how scraggly Shaggy looks. It seems like they're struggling to get by for the most part.
5. The gang actually is around high-school age: between 15 and 17. They might look older, but their official age as given in the show bible were given as Fred and Shaggy being 17, Daphne being 16 and Velma being 15. So if these kids are of high-school age, why are they not at school? It seems like these are just a bunch of bored kids that have nothing better to do with their time than to drive around the country in search of things to do.

All of this evidence points to Scooby Doo, and especially the original 2 seasons, to be set in the middle of an economic recession. Most people with specialized skills don't have jobs, and have turned to crime in hopes of getting money. Tourism and education have gone down the drain because people don't have the money to travel or pay for education.

So they might simply not need that much money because there isn't that much to spend it on. The gang may just simply be a roaming group of high school age teenagers that are in search of anything to do.

According to an interview of Casey Kasem that I read, the entire Mystery Inc. team is wealthy.

Scooby is blue blood old money. His family is pedigrees champion stock. But most of his money came from an eccentric dutchess who left her entire estate in trust to Scooby's family. As such, with all assets tallied, he is the wealthiest.

Norville "Shaggy" Rodgers is also blue blood old money. His family owns several estates, castles, etc. in Europe. They also own vast quantities of land in the US, and own or have controlling interest in many businesses.

Scooby and Shaggy were caricatures of students attending Brown and Princeton Universities at the time the original show premiered.

Velma is old money, but from a more salt of the earth stance. Her family has owned bookstores for generations, again going back to Europe. The specialized in rare and antique books. They did suffer numerous financial setbacks due to various issues, but were always able to persevere and rebuild. This was due to, (and also reinforced), their down to earth nature and work harder and smarter mentality.

Velma represented M.I.T.

Daphne's family is new old money. Their wealth came from railroads and industry in the US.

Daphne represented Yale.

Freddie's family is politics money. Like Daphne's family, they made their money after the US was founded. Compared to the rest of the gang, Freddie's family would be considered new money, as they have only been wealthy for 5-6 generations or so. (Post civil war)

Fred represents Harvard.

Freddie was the "fearless jock" and defacto leader. He did have a knack for traps.

Velma was the tech guru with a penchant for forensics, investigating.

Daphne was the "face" of the gang. Skilled at persuasion and snooping, she still frequently fell into the role of resident "Damsel in distress".

Shaggy was the undercover mech genius. While Freddie may have done most of the driving, Shaggy owned, maintained, and enhanced/upgraded the Mystery Machine. He collaborated with Velma in building their mobile crime computer/lab, and helped Fred with trap construction more than any other member of the gang.

Scooby was of course the lovable goofball mascot, as well as the gang's "bloodhound" for sniffing out clues.

During the original series from the late 60's and early 70's, the whole gang were trust fund kids who didn't need to work. Though they often took on odd jobs as part of their investigations. (That money, if any, most likely supplemented the food expenses.)

[During the late 70's and on, their previous experiences helped mold their growth into individuals, and led them to their own chosen professions with the dissolution of Mystery Inc. until they were reunited and chose to revive their detective agency.]<---this is my personal opinion and in no way derived from the interview I read. I also chose not to go into the darker side of the gang's alleged partaking of recreational pharmaceuticals.

They all have rich parents and don't travel far, so considering gas prices are low in the US, it is very plausible that they use their allowance to chip in for gas and that is all they need. Even Scooby-Doo comes from a rich dog family, and Shaggy received some big plantation as an inheritance. Only Velma is middle-class, but she worked for her parents in a museum.

It's important to note that Scooby-Doo is not one unified continuity and that how they got financed if at all. It's also worth noting that the show has generally gone back and forth focusing on mystery and comedy (especially in the 70s and 80s) and was somewhat revived as a mystery with side of comedy in the late 90s. Usually the post 90s is shown that the original What's New Scooby-Doo and First Scrappy-Doo seasons are somewhat in the same continuity but this doesn't include the 70s and 80s especially if the series featued part of the five heroes (The Scrappy-Doo era is often less discussed, but early episodes of him did feature mysteries that were in line with original series). Generally the Scooby-Doo movies, and the adventures featuring Only Shaggy, Velma (dropped later) Scooby and Scrappy are not cannon, nor is a "Pup Named Scooby-Doo" though it was considered better.

Scooby-Doo was revived in 1998 with the direct to video release "Scooby-Doo on Zombie Island" which billed that "this time the Monsters are Real" (despite most of the comedy films already doing that... though this was more inline with classic Scooby-Doo) and featured the gang as adults in their 20s with Fred and Daphne working on a travel show called "Coast to Coast with Daphene Blake" with Daphne staring and Fred Producing. The intro scene reveals that by this point, the Mystery Inc gang known well enough for their mystery solving that an Operah like talk show host discusses their adventures. At the same time, Shaggy is a recently fired customs (for eating all the contraband) agent with Scooby as his canine partner and Velma owns a book shop and Fred hires the three on as part of the tvshow's research staff as the new season will focus on "Haunted America", thus explaining the set up for the later direct to video films and the "What's New Scooby-Doo tv series" and featured some characters from the films coming into the show (Notably the band "The Hex Girls" first introduced in the second film "The Witches Ghost" and were fan favorites).

From this, there were a few consistent elements of financial sources:

• Daphne, Shaggy, and Fred were all from well off families while Velma's seemed to be a bit more working class, fitting her more down to earth characterization. While Daphne and Fred's family were proud of their child, Shaggy's laid back attitude was a source of embaressment and his family was more supportive of his activities because it meant not having him around the house. No sources of income were depicted though all wealthy families were depicted as "old money".

• The Mystery Machine was Fred's car, and Scooby-Doo was Shaggy's dog. In "A Pup Named Scooby-Doo, Scooby's parents were wealthy dogs as well. Although this was dropped, it's possible that his parents played famous TV show dogs (ala Lassie) and Shaggy's family got Scooby from a liter for Shaggy (personal theory only). Another good theory is that Scooby's family paid them a hefty sum to put up with Scrappy.

• The town they grew up in was most likely on the U.S. South East and was a summer vacation town as it had a beaches (with Yacht club), swamps, farms, and one original episode depicted a Confederate Mansion. It is possible that they traveled over a wider area and an alternative location could be California as prospecting history and Asian immigrants were depicted in the original series and the beaches were not in line with those on the East Coast. San Francisco was one of the earliest real world places featured and when given, the town's name is said to be "Coolsville".

• From 1998 revival onward, they seem to have had a series of jobs and may be working for a television program or journalist piece. The gang's exploits are known to the wider public as an Operah like figure was more interested in Daphene's stories about the good ol' days then the TV show she was plugging, and Ben Ravenscroft, a famous horror writer does seek them out specifically in another movie. The following films didn't use this angle nor did the "What's New" series, though both depicted much wider international traveling which suggest this could be there funding source.

From here, the series returned breifly to a Shaggy and Scooby only comedy before getting rebooted into the much praised Scooby-Doo: Mystery Inc. which actually filled in alot of gap economic information. This time, the series was explicitly set in the beach town Coolsville, California. The local economy survives on tourism since the "Alligator Mines" dried up (yes you read that right... It's probably the weirdest thing in the show... and the series featured a Lovecraft-esque writer trying to write his cosmic horror characters into a teen romance after a Stephanie Meyers parody mocks him for not writing what people want to read, so yeah, it's pretty weird). One of the big draws aside from the beach is the town's reputation as one of the most haunted locations in the nation... which causes the gang no end of grief because they are always solving the mysteries of the ghosts, thus ruining the possible new attraction. It also lends cover to the masked criminals (in one case the guy in the mask wasn't a criminal as none of his deception was an actual crime) as one more Ghost is expected and the local areas ready belief in non-rational explination as the town not only is motivated financially to go with the new ghost story, but also because there are some strange things about the town that would suggest there are some real ghosts and monsters. All the kids families now have a vested interest in the haunted tourism industry as they are either invested in the industry or the town's success and their own children are all embarrassing them by proving the ghosts aren't real.

The gang are also teenagers and this time their school life and family lives are fleshed out a bit more as well. Shaggy and Daphne retain their old money family roots, though this time Daphne's family also do not approve of her mystery solving hobbies and to a lesser degree, Fred. Their motive is that Daphne is the youngest of four or five sisters, all of whom are successful and all whom went to the same elite university their parents met at and clash with Daphne, who wants to be less identical. Shaggy's family has their traditional depictions of disapproval, but it's much more openly depicted. Velma's parents finally are given a job, here working class, as they are the curators the town's ghost museum, which Velma works as a tour guide after schools, though they quickly decide not to make her do this, as her parents leave out the fact that the ghosts were all proven hoaxes and Velma will rattle off the crimes of various ghosts (all from the original series plots) to any tour group.

The biggest difference is Fred, who is not from an Old Money family but rather his single father is the Mayor of Coolsville. As the duly elected representative of a town that hates both crime and revealing ghosts to be criminals, Fred's dad is also not in favor of his son's hobbies, which aren't helped by the fact that Fred in this continuity has some issues grasping social norms (he loves making traps... which is weird but... not to an alligator mine level. At least even his friends think he's a bit too obsessed). It's also quickly shown that Mr. Jones may not be the honest politician one would expect, given the town's tendency to look the other way whenever a ghost shows up and he has some knowledge of the overarching mystery presented in the series lore.

As far as continuity, it's a toss up, as the show all but says the original series ghosts all happened in Coolsville and were all solved by the gang. The Ghost Museum additionally features an exhibit dedicated to the character Flim-Flam (from the 70s-80s comedy era) who Daphne wonders what happened to him and Scrappy-Doo, which Daphne shrieks in terror when she see's his statue and when she tries to apologize and explain why she reacted like she did, Fred quickly reminds her that the entire gang swore they would never discuss the events surrounding Scrappy again. Given the show's blend of being way more darker and edgier than typical Scooby-Doo, but still paying a loving respect to the series invokes, it's hard to tell what previous were cannon, and as the series progressed, events would further throw this into doubt. However, given that the series was widely loved by long term fans, expect that the characterization of the families and some other popular characters would be reflective of this depiction (this particular depiction was homaged in an episode of Young Justice's season 3.).