Hello, Melisa,
Rest-assured, Walt Disney World does an tremendous job accommodating dietary restrictions. I don't have a specific list of snacks that are both dairy and egg free, but if you contact Disney at SpecialDiets@DisneyWorld.com I think they should be able to help you with this.
I'm also going to link you to information on
Special Dietary Needs that includes a list of quick service restaurants (toward the bottom of the page) that should be to accommodate your son's needs upon request.
And I have to tell you that there is a small bakery located in
Pollo Campero at Downtown Disney. BabyCakes NYC specializes in baked goods that have no gluten, dairy and eggs. We sampled just a few weeks ago. May I say, "yummy!"
Have a great trip and please let us know if we can help again.