Disney is wonderful for people with food allergies! My DS is severely allergic to eggs, and can (now, thankfully) have very limited dairy (like if it's in a bread now he can tolerate a small amount, he's outgrowing it now). When we first took him to Disney at 3, he was also allergic to sesame and peanuts (he outgrew those, and I'm glad we get his blood tested annually). I am glad about the dairy, as Wonder Bread a few years ago started adding dairy to ALL their breads. I forgot to read the label (you have to each time you buy something, as they change ingredients occasionally) and he reacted to buns once. UGH. Well, most of Disney's buns come from Interstate bakery, which makes Wonder bread. So if you avoid dairy, find out about the buns at the counter service places. They do have kaiser rolls sometimes that are dairy and egg free.
The counter service places usually have a binder with ingredients of all their foods. Just ask when ordering. They have been so helpful with substitutions for my DS. The only places to be cautious seem to be some of the country pavilions in Epcot world showcase. At the table service restaurants, mention food allergy when making a reservation. They will have a chef come speak to you personally at your table. My son was thrilled that they could make him so many things. When he was young, we would still sometimes order him adult items. We were so thrilled to have him be able to enjoy so many great meals.
We've been many places with our kids, but I keep loving to go back to Disney. This is one draw for me - the care they place on dietary issues.
We did Universal once, and stayed at the Portofino Bay Hotel. They were not helpful at all with dietary needs. I would rather not go back there again.
It is helpful also to look at menus at allears.net. That can help give an idea of the types of options you'll have.