Yes, yes, and yes - WDW can handle Celiac very well....
Our DD has Celiac - gluten free and never had a problem at WDW
By the way - the popcorn at the snack wagons is gluten free, and so are the Ikadozzie ice pops (how ever you spell it, our DD's favorite).
All sit-down places can easily accomodate gluten free - just mention it on your ADR and call one of the #'s below to connect to the specific restautant.
You can always get roasted/bar-b-qued chicken (no teriyaki sauce), bunless hamburgers, salads (no crotons), fruit cups, and ask about the french fries - if they are flour coated and fried separately.
Toy Story pizza in DHS has gluten free pizza (just ask- takes 10 minutes to prepare)
Website with Gluten Free Guest comments
http://allears.net/din/guestceliac.htm
And ....
If you have specific inquiries or would like recommendations regarding the food served at the park restaurants, you are welcome to call the executive chefs or Food and Beverage managers for our theme parks, at the telephone numbers listed below. If no one is immediately available when you call, please be sure to leave a message and a telephone number where you may be reached.
Magic Kingdom Park: 407-824-5967
Epcot: 407-560-7292
Disney-MGM Studios: 407-560-1347
Disney's Animal Kingdom: 407-939-7536 (Please note Walt Disney World Company operates no full-service restaurants in this park.)