My son is also allergic to dairy and egg. He's had anaphylaxis to dairy, and when he was younger he was touch hypersensitive (he'd hive up touching anywhere that dairy had been, like restaurant tables). He luckily outgrew peanut and sesame!
Disney is the best place to vacation for food allergies, especially the table service places (or any counter service other than maybe a few in Epcot World Showcase - but he loves Japanese chicken teriyaki and rice, DH is Korean so DS grew up eating rice all the time, like daily when he was a baby since it was safe). At TS they will bring a chef out to talk to you. CS has ingredient lists, and in the past they would make substitutions for him. At Restaurantasaurus in the AK, they even remembered us from one day to the next! Subbed out a hot dog with no bun for the kids' meal option, and they also will change gloves (saw it on his ticket for his meal "change gloves"). Awesome.
We always did have snacks in the room, and soy milk (now we're doing almond more for him). Garden Grocer has a good selection, but you can also use
Amazon Prime Pantry, Amazon Prime Now, or other options (I always had snacks in my carry-on for him, since it's so tough to travel long with allergies - try flying 10 hours or so to Hawaii, when what they sell on the plane in coach isn't safe - ugh).
I am sure you'll be fine!
Check out the disabilities board here!!!!