We had no issues, they are not allowed to ask your disability.
I'm not exactly sure how this belief became so widespread, because it is simply not true. Businesses, whether theme parks or movie theaters, are not required by law to "take your word for it" regarding your disability. They can require proof of need of an accommodation. Of course, most don't, because most accommodations are simple enough to meet, and it isn't really customer-friendly or worth the hassle. Funnily enough, the only thing businesses can't require is proof that a service animal is in fact a service animal.