My DS is dairy and egg allergic (he has had anaphylaxis from dairy and even reacted for the first time to his flu shot this past winter due to his egg allergy, even though his numbers for both have dropped dramatically over the last few years). So I can relate!
Disney is the best vacation spot we've ever gone for his food allergies!
All the buns, hotdog and hamburger, are made by Interstate Bakery, which makes Wonder Bread. Unfortunately, about 6 years ago they started adding dairy to all their breads. So at a counter service place, make sure you get dairy free breads! They frequently will have kaiser rolls to substitute that are dairy free. He's eaten hot dogs without the bun, cut up of course for a 2 year old, as they didn't have any other buns.
If you go to a table service place, make a note of the allergy. He's gotten Rice Dream and Tofutti before, and felt so special to get that!
The table service places will help immensely with choices, and the counter service ones have books with ingredients as well. We had to have a cast member call a manager once to find out about the pretzels (I felt bad, he went to so much trouble, as his cart didn't have the ingredients) and did find out that at least on our last trip, the pretzels were fine (we really need to make sure, as one time at a restaurant in Baltimore, the pretzels had egg wash, and I didn't ask, he reacted so bad, took 3 or 4 teaspoons of Benedryl). Still, I'd ask each trip to make sure there aren't changes.
We used to also bring cookies and things, so he could have a treat.
We've been going to Johns Hopkins for allergy testing since he was a baby (first diagnosed after anaphylaxis at 8 months at an army hospital), and get blood tests once a year. That has helped, as he outgrew sesame and peanut. The test is called a RAST. I really recommend it, if you can. My DD outgrew her milk allergy at age two, luckily.
Good luck!