Which Dungeons & Dragons character class does Max belong to?

Based on her actions and abilities we've witnessed and based on the time period setting of the series, I'd suggest she definitely fits the Thief/Rogue archetype. From the linked Wikipedia description,

The abilities of the thief class were drawn from various archetypes from history and myth, but clear debts from modern fantasy literature can be traced to characters such as J.R.R. Tolkien's Bilbo Baggins, Fritz Leiber's The Gray Mouser, and Jack Vance's Cugel the Clever.

and

In 2nd edition the term "Rogue" first appeared, used to describe the group of classes made up of those individuals "living by their wits day to day-often at the expense of others." In the core rules, these "rogue" classes were the thief and the bard. Thieves could be of any alignment other than lawful good while bards had to be at least partially neutral.

Being the "new kid" in a relatively tight-knit community, and having a violent and unpredictable older step-brother who treats her with, at best, contempt, it seems to me that Max is forced to "live by her wits" from her first day in Hawkins, IN. Also, as has been mentioned in the comments by @ElDuderino and @Phlyk, she seems to exhibit abilities and proficiencies that would be indicative of the Thief/Rogue class. Perhaps it's a bit of a stretch, but something akin to the "Trap Sense" ability (for example) is evident early on when she quickly picks up on the fact that the rest of the Party is following and watching her.

Besides, the Party needs a Thief/Rogue in the same way that Thorin and Company need a burglar at the beginning of J.R.R Tolkien's "The Hobbit". An interesting point of view from one of the answers to this question on Quora (The Hobbit (book, movies, and creative franchise): Why did the quest need a burglar?) says this:

[Thorin and Company] didn't really need burglar, it needed Bilbo Baggins IF Gandalf was to go along with the dwarves and aid them in their quest. It was a condition upon which Gandalf's agreement depended.

Gandalf had a feeling about Bilbo, a sort of sixth sense that told him that it was important for Bilbo to go. He didn't know why until later, when he believed that Bilbo had to go because Bilbo was MEANT to find the One Ring and Frodo was MEANT to have it.

Thorin and company were not happy having an overweight, cowardly(as they thought him) hobbit tagging along with them; but the desire for the wizard Gandalf's aid outweighed his objections.

In the end, it was a good thing for them, as Bilbo saved their lives on several occassions.

However, also like Thorin Oakenshield and many of the dwarves in his Company, the leader of the Party in Stranger Things, Mike, doesn't immediately recognize their need for this character class. Unlike the hesitant Hobbit, however, Max actually wants to be included in the Party. Quoted from the show [S02E03 - Chapter 3: The Pollywog] where Mike is arguing with Max that they don't need another party member:

Max: Why do you hate me so much?

Mike: I don't hate you. How can I hate you? I don't know you.

Max: But you don't want me in your party.

Mike: Correct.

Max: Why not?

Mike: Because you're annoying. Also, we don't need another party member. I’m our paladin, Will’s our cleric, Dustin’s our bard, Lucas is our ranger, and El is our mage.

Max: El? Who's El?

Mike: Someone. No one.

Max: Someone or no one?

Mike: She was in our party a long time ago. She moved away, okay?

Max: She was a mage? Well, what could she do? Like magic tricks or something? Well, I could be your zoomer.

Mike: That's not even a real thing.

Max: It could be.

EDIT: I found this Tumblr post with some descriptive animated GIF's for the Party's classes. It includes the non-class "Zoomer" that Max suggests in her conversation with Mike, but otherwise, it's a fairly decent representation.

http://becamitchill.tumblr.com/post/167067168036/paladin-wheeler-stranger-things-dungeons-and