Soy-free and gluten-free makes it nearly impossible, especially if you have to avoid soybean oil. Baja Fresh is okay in some areas, but you have to ask for ingredients and fresh grill space/foil. You might tolerate Five Guys (lettuce-wrapped burger, fries are in peanut oil) *if* the staff are scrupulous about using clean grill space, clean hands, fresh ingredients, etc.
Disney might be able to accommodate you with prior notice.
I carry food everywhere. We travel with our camp kitchen and food (cooler, bins) and cook out when the weather is agreeable. Otherwise, we find hotels with microwaves and cook veggies, microwave brown rice, and subsist on safe lunchmeats. I make my own bread to take. Or live off corn thins and sliced cheese.