When did Spider-Man get his mechanical web thrower?

In the comics, way before he could naturally shoot webs; in the movies, he always uses them, or never uses them - it depends what movie your watching.

In the original Amazing Spider-Man comics, web shooters were a device Peter Parker created, in order to complement his new-found abilities; his ability to shoot webs are almost always a result of this device.

The Sam Raimi trilogy, the three starring Tobey Maguire, was based on a script that was initially written by James Cameron. James Cameron had the brilliant idea that "organic web shooters" made more sense, and thus depicted them as such. This would later be adapted into the comic books, where in Peter Parker goes through more transformation, and grows organic web shooters; however, in the movie storyline, Spider-Man is given organic web shooters from the start. Regardless, in almost all depictions, the web-shooters are actual gadgets.

The gadget web-shooters make a return in 2012's The Amazing Spider-Man. Once again, this is a different universe to the Tobey Maguire Spider-Man and Homecoming. In Homecoming, this device is created by Peter and improved by Tony Stark¹. Again, this is a completely differant universe - the "Marvel Cinematic Universe". It has no canonical connection to The Amazing Spider-Man movies, or the Spider-Man movies created by Sam Raimi.

_{¹ Thanks to cde for clearing that up.}

Spider-Man has normally always had mechanical web shooters. Since he was created as a comic character, and in multiple cartoons and movies. The organic web shooters were invented in 2002 as a reimagining of Spider-Man by director Sam Rami. See http://www.cbr.com/follow-the-path-spider-mans-organic-webshooters/ This was later implemented in the Spider-Man comics in 2004 So it would match the current movie version of Spider-Man at the time the series was made. It was a highly contentious change, for both the movie and comics. Marvel has since implemented many changes to their comics to follow the movies (Spiderman and most of the Avengers in the comics now resemble their movie counterparts, introduction of movie only characters like Coulson into the comic universe).

The Amazing Spider-Man trilogy and Homecoming go back to the original design where Peter Parker uses Mechanical web shooters. The biological ones are the odd man out. Each of these movie series are a different take on Spider-Man, and are not in continuity with each other.

Homecoming implements part of a different take on Spider-Man, where the comic event Civil War included the Iron Spider. This version had Peter Parker use a suit created by Tony Stark, which included all sorts of Iron Man/StarkTech upgrades. Tony gives it to Parker because Parker sides with Tony's pro-registration side of super heroes. Much like he does in the movie Captain America, Civil War, which adapts the Iron Man vs Capt. Storyline of the comic event of the same name.