I think you will be fine! The best options will be at sit down restaurants as well as packing some safe snacks to have with you, both when you're out at the parks as well as in the room.
My DS has diary and egg allergy, but also used to have peanut and sesame (he outgrew those by age 6). He gets tested every years with a blood test at Johns Hopkins hospital, and his levels of egg and dairy continue to drop, but still not low enough (and he had a reaction to his flu shot this year, yikes).
At a sit down restaurant, a chef will come to your table to talk to you or the child, and see what they can make. Sometimes for my son it would be plain grilled fish or chicken, or steak, plain rice or some other potato, veggies, rice dream or tofutti, etc... The restaurants in the countries in Epcot are a bit different, in that we don't even bother trying a place like Italy - not sure if the pasta would have egg, so it's really not worth it to try (there are so many other choices of places he loves). Upstairs in France, at the Bistro - we had an over the top (and $$$$) meal - he had a lovely filet of beef, salmon appetizer, so much great food.
Even last March, we stopped by Captain Cook's at the Poly for a lunch one afternoon, and my son wanted the turkey sandwich. The manager came over to talk to me, since the bread wasn't safe, and it normally came with cheese, etc... She made him a plain turkey sandwich, and he wanted fries. She said the fries were cooked in shared oil with the chicken nuggets, so she had an option of "allergy fries" which were totally safe. Took a few extra minutes, but she was so careful! I think the fries were just plain potatoes cooked in their own oil - not shared.
I definitely always felt safe bringing a small bag with some safe snacks along, just in case he got hungry. At Disneyworld, though, you will find better care for allergies than anywhere we've ever been! That is one reason we've gone so many times, and I started taking my kids even without my husband when they were 4 and 6 - I felt so safe!