First check the FAQs that Cheshire Figment mentioned as it has lots of links for people with special diets.
Contact Bennett at WDW via email. They strongly recommend getting the latest version of their list of allergens and stuff. New information comes to light or products and recipes can change.
Table service restaurants are excellent for preparing vegan, diabetic, allegen free and other special diets. In FAQs you will find links to reports on how good the service usually is. The only restaurants I have heard bad stuff about is the new restaurant run by Landry's near Everest in Asia and the Mexican restaurant in Epcot. Just have it noted on dining reservations and they will be ready for you when you show up. Is there a best retaurant to eat at if you are GF? It depends on what you like to eat as there is really very little a gluten free person cannot eat while at WDW.
Counter service restaurants offer gluten free hot dog and hamburger buns. It takes planning and they have a giant binder listing all the allergens. This is at each restaurant.
The chefs walk special diet guests through the buffet lines, will make special meals, will serve buffet items from new batches, and will take time to discuss the menu. They try to avoid all cross contamination like by using clean salad mixing bowls instead of the ones that have croutons in them.