Disney is fantastic at accommodating food allergies. As long as you identify the allergy at each and every place you eat and each and every time you place an order, restaurants will be able to accommodate them. We avoid buffets, but if you go to one, be sure to ask the chef to make your plate in the kitchen so there is no cross-contamination. Chefs are excellent at making safe meals or modifying existing menu items to be safe (making a special plate from scratch).
At quick service locations be sure to mention the allergy (every time) and ask for the manager to bring out the allergy binder. They have a list of ingredients for everything at every food service location. Some quick service locations have special allergy menus and meal options. Sunshine Seasons in the Land pavillion at Epcot has a big selection of safe treats too.
Animal Kingdom has a special diets kiosk in the park with snacks suitable for people with various food allergies and sensitivities. I haven't seen it yet but I've heard they have a wide assortment of safe snacks.
You may be surprised at how many safe foods there are at Disney. We even found Enjoy Life cookies at our hotel. YAY! The chef at our hotel was fantastic too. After meeting him the first night of our trip, he brought in extra allergy-safe foods so I had a wider variety of meals to choose from each day. Allergy friendly Mickey Waffles were such a treat!
Babycakes in Downtown Disney should be safe too - lots of allergy-friendly desserts there!
Always be careful, carry your autoinjectors (EpiPen, Allerject, etc.) everywhere you go, check ingredients on everything, and have fun!