Hi Emily,
I think you will find that Disney will be very understanding and accommodating to your daughter's allergies. You certainly can bring your own food with you, and you can ask to speak to a chef or manager in any Disney restaurant to discuss ingredients, preparation and adjustments to menu items for allergies.
The
special dietary requests page on the Disney website will give you more details.
Disney is looking at a number of innovations to help guests like your daughter. At the new Be Our Guest restaurant at Magic Kingdom, guests use a touch-screen to order lunch and are able to note allergies before they begin. When you check that you have an allergy, any items that could be harmful disappear from the screen.
You also are welcome to bring in your daughter's epi-pen or any medication that may be necessary for her. It is suggested that you tell the cast members at the security checkpoint that you have an epi-pen, medications or foods for special dietary needs.
I hope that eases your mind. Please feel free to come back to the me or any other member of the Moms Panel if you have more planning questions. Enjoy your time with your family at Disney!
Wilma