Someone on the Gen Con staff can clarify or correct this information as needed.
I don't see any age restrictions on opening an account or buying a badge. BUT, from the Attendee Guide:
"The majority of Indianapolis hoteliers have raised their minimum guest age check-in requirements from 18 to 21 years of age. To ensure guest compliance and hotel accommodations, Management recommends that registrants under 21 either refrain from booking rooms through the housing portal or review hotel policies carefully to confirm guest age sufficiency before hotel selection. Registrants are also encouraged to carry a photo ID to present at check-in."
https://www.gencon.com/gen-con-indy/housing_travel
So if you want to create an account or just buy a badge and nothing else, I don't see an age requirement. There might be one, but I haven't found it.
If you want to do that and get a hotel, then the effective restriction imposed by the hotels (not Gen Con!) is 21 years of age. That of course is not enforced by Gen Con, but obviously you'd risk running into problems at the hotel itself. Depending on it's policy.
Again: Subject to correction by the Gen Con staff. If Mike Boozer or anyone else from there replies, listen to them!