My son has a dairy allergy and I have taken him 3 times plus 2 lunches in the then downtown Disney on the way to/from the auto train.
He ate everywhere with us. I did bring his butter, earth balance, because at the time Disney just had fleishmans unsalted with has no flavor. Plus I wanted to make him Rice Krispie treats.
The only problem we encountered was in the then downtown Disney and we were going to eat at Rosie's all American cafe or whatever it is called. I forgot his burger buns, I brought them too because I knew he wouldn't eat the gluten free ones and they were unable to accommodate.
And when we ate at Wolfgang pucks express, a life time ago, 2013, in April, which was a pit stop off the auto train, he ate the pizza with out cheese but we put on pepperoni. We went to Disney that following August and planned on eating there again and they told us we were lucky because In about 2 weeks their dough was going to have dairy in it. So sad. We have been bummed but...December of 2015 he was given the all clear to eat baked milk so we were thrilled he could eat there and even better, 2 weeks ago another food challenge and he can eat cooked cheese so I don't have to spend an arm and a leg on a dairy free pizza in Disney with soy cheese!
Anyways, he also has an egg white allergy (he eats eggs cooked in things), peanut (failed the food challenge 3 yrs ago) and most tree nuts but he can safely eat almonds and cashews as of a year ago. So yes he was able to eat everywhere and we weren't just dealing with dairy.
We are not big TS eaters, 2 per trip, and we ate at CRT for breakfast, ohana for breakfast (both a waste since he only likes the bacon and sausage, won't eat a pancake/waffle, he is 13 and I forced him to take a bite of one 2 weeks ago and he said nope I stilll hate them), Whispering Canyon cafe, crystal palace and chef mickeys. All others have been CS.