Some restaurants have dedicated fryers in which only french fries are cooked. All other fried food such as cheese, zuchinni, shrimp, and onion rings are cooked in the same oil. The only way to prevent cross contamination of deep fryed foods is eat at a restaurant that does not serve that food.
I cannot eat french fries cooked in a common fryer because of the soy and wheat in onion rings, breaded meats, and breaded veggies. As stated above read FAQs for information on contacting WDW. They can help find allergy free foods. A sit down restaurant's chef might make your child some deep fried cheese by hand instead of a deep fryer for example. Counter restaurants are less flexible and you have to make sure you let them know of any allergies. This site has lists of menus that you will need to read so you will know in advance if they also serve seafood for example. Cross contamination can also come from cookware, utensils, and counter tops for example. Soy seems to be in everything and I am soy sensitive.