Perhaps slightly off topic, but by far the most difficult thing of my entire Disney career was dealing with GAC cards. Boy, talk about a sensitive issue! The problem was, in my experience, is that their usage is out of control. Now keep in mind I only saw a fraction of guests in a day who actually entered the park, so to me there was a proportionally large size of people wanting GACs. The apprehensiveness of CMs, as someone stated above, has nothing to do with the genuine guests with disabilities. You're just talking about a heavily abused system which also has millions of combinations of variables. What I mean by that is that some people think they NEED a GAC for one thing while someone else may just go about their day without it.
Like many things, the issuance of GACs is a "judgement call" (I hate that term!). It's difficult because you could really help out someone who needs it or you could really mess up someone's trip by denying it.
And consider, again, all those variables! What if a parent comes in saying they have an autistic kid they need a GAC for and they have a party of 10, but the kid is no where to be found? What do you do? Ask to see the kid? But, wait, autism isn't readily apparent so even when they bring in the kid, what are you proving?
What about "adult ADHD"? GAC?
Someone already in a wheelchair? Do they need a GAC?
The issue is extremely sensitive but my advice is simple: go to Guest Relations. Explain things just as you did here, and ask "Is there any assistance you can provide me?" The CM will then explain not only the GAC but all the other wonderful ways Disney goes out of their way to especially help those with disabilities. By no mean sshould you feel strange about using it once you have it; honestly the CMs at the attractions see literally hundreds of these a day; no one is "checking you out" to see if you're really disabled or something.
I'm sure it must be extremely difficult to be put in that position. All I can do is explain my issues, show my medicines (photocopy with date and my name) show my handicapped plate, and doctor's note. I can actually show one part of the disability that's easy to see... the other's... not so much. I personally think it's very difficult to ask for help. Part of me wants to hang on as long as I can without one... but, the other part had a very difficult time last year and I know I can't go on the rides without it.