First, when you make reservations, make sure the allergy is noted on each one. Also, ask them for the contact information for the person in each of the parks who handles guests with food allergies. They will help you determine what can be done. I have a friend with a wheat allergy and each park provided her with a list of all the carts and counter service locations that served something gluten free. I imagine they can help you, too.
In general, sit-down restaurants are very good at working with food allergies. You will be given the contact information for each restaurant and you can call 48 hours before your meal and speak with the chef. They will cook seperate batches of food in cleaned pans to avoid cross-contamination. If you need the pan bleached or double washed, I think they would do that. They really take food allergies seriously.