No, you don't need to restrict yourself to shows for riders with disabilities, you can compete anywhere.
If you wanted to compete at recognized shows using adaptive equipment, you can apply to the USEF for a "dispensation certificate" that will allow you to use your special equipment and specifies what that equipment is. (You didn't mention what sport you compete in, but the USDF has a good FAQ page about competing in recognized dressage shows here. The same principles about the dispensation certificate holds true in other USEF-recog shows.)
If you are just interested in schooling shows/barn shows, those are typically informal enough that it should only require a call to the organizers to tell them what you want and have them make a ruling on whether it's acceptable to them. Even then, though, I would expect they'd follow the precedent set by the USEF to allow people with recognized disabilities to use certain compensatory aides or tack.
I would guess that, if you expect to compete at recognized shows at some point in the near future, it might be worth going through the process of getting the dispensation certificate. I can't imagine a local show organizer arguing against allowing you to use something the USEF has already allowed, but they might think your situation is more open for debate if you haven't already gotten that approval.
What kinds of special equipment do you want to use?