Yes. I don't think they have as much leeway for providing alternative (I've never heard of them having soy dream or rice dream, for instance), but they certainly do not want people to become ill.
Call Disney Dining, ask to speak to the chef in charge of food allergies. You should be put through to voicemail; leave a detailed message with your name and phone number and the problems, and they should call you back asap to help.
When I called, with a question about a rumor involving vegetarian items (unfounded rumor; if it says vegetarian it IS vegetarian), I spend an hour talking with Chef Chris, and they really do care. They had even made a raw lasagna for a raw foodist, at one of the hotel dining places, a few weeks before I called.
Then make PSs with the allergies noted, and remind the server.