I am also the parent of a child with a very severe egg allergy. My DD is almost 7, so we've been dealing with this for a while. We love traveling to WDW because they are so accommodating.
You will have plenty of options at the table service meals. The chefs are very knowledgable regarding ingredients and have allergy friendly options available.
One thing to be careful about at buffets is cross contamination. I usually ask the chef to bring out my DD's food from the kitchen because if someone used, for example, the fresh fruit serving spoon on a salad with a mayo based dressing, it could be a big problem for DD.
At CS restaurants, there are binders at the cash registers with ingredient listings. As of last year, the CS mac and cheese had eggs in it, but they had Amy's mac and cheese available as an alternative.