A good deal of the counter service locations have gluten-free food options. Like someone suggested, e-mail Brenda and she'll send a list of what you can get and where.
My sister, who has celiac, recently ate at Cosmic Ray's in the Magic Kingdom and got a double cheeseburger with the gluten-free buns. She said they are a lot better than they used to be and they almost taste normal.
We split a hot dog at Liberty Tree, which is one of the best gluten-free options. That is in Epcot. They have dedicated fryers, so he can get french fries, without any cross-contamination. The gluten-free bun that came with the hot dog was not bad (and this is coming from someone who doesn't need to eat gluten-free). They also have gluten-free brownies and cookies.
In the past, we have eaten at Toy Story Pizza Planet and they have Amy's Pizza there that they can prepare for you. I believe they also have that at Pinocchio Village Haus.
At Animal Kingdom, your choices are pretty limited and it's basically only Flame Tree BBQ that you can eat at.
In terms of a couple resorts, Chef Jenny at the Mara at Animal Kingdom Lodge was great and told my sister she would prepare her whatever she wanted. She made her rotisserie chicken with rice (since cous-cous was not ok for her) and then another day made her ham and cheese sandwiches on gluten-free buns. At Saratoga Springs, Chef Ron is always amazing and is willing to go above and beyond to make things special. My sister likes to go to Arist Palette and get gluten-free buffalo chicken strips. He makes them special for her with rice flour. He is VERY nice. Be sure to ask for him, as not all the chefs there are willing to do anything more than just tell you what you can and cannot have, without accomodating your allergy.
I hope this was a little helpful. For the best info, I would really contact Brenda. If you have any more questions, feel free to send me a PM.
