No, there is no official "Adventure Zone D&D Ruleset"
They haven't described the changes they've made to the rules, because that would be boring to most people.
I have listened to every episode, some multiple times (it's my favorite podcast). Griffin has never mentioned a specific set of rules, nor are they listed anywhere on their Maximum Fun page or their Tumblr.
The reason for this is that they want the show to be entertaining for people who don't play RPG's, it's a comedy podcast first, a D&D podcast second. They don't want to be bound to any rules, even of their own making, because don't want rules to get in the way of a fun show.
There are a few rules that they seem to have decided beforehand to ignore, like XP (they generally level up their characters after they complete a story arc) and inventory management (they don't worry about encumbrance or buying uninteresting things like trail rations, bedrolls, etc.) Keeping track of these sorts of things would be very boring for most people. Also, Griffin has made up most of their magic items to fit the setting and to be interesting.
Other than that, Griffin makes rule decisions on the fly based on what would make the most entertaining show. They rarely refer to the rule book (except when Merl or Taako cast a spell they've never used before) because that would be boring for most people.
Keep in mind a significant portion of the listeners of the show don't care about rules, they listen to the show because they are fans of the McElroy brothers.
That being said, apart from the things I listed above, I haven't noticed many significant changes to the rules. I'm sure someone more familiar with 5e could point out a few more differences, but as far as I can tell the game they play isn't hugely different from RAW.
Update: In the episode The The Adventure Zone Zone the cast discusses the show and answers some listener questions, much of their discussion is relevant to this topic. They do not spell out a set of rules, but they talk about how they decide what rules to follow and why they do it that way.