Victoria,
My niece has severe allergies to tree nuts, dairy, and eggs, so this is always a concern when our families travel to Walt Disney World together. We have ALWAYS had great success in our Disney Dining experiences. In fact, Disney is very committed to doing their best to accommodate the needs of Guests with food allergies. The key is planning ahead.
If/when you make Disney Dining reservations in advance, always inform the Cast Member taking your reservations of the allergies. They will make a notation on the reservation. In addition, always tell the Cast Member seating you about the allergies and ask them to inform the chef. In many instances, you can speak directly to the chef and let him/her know of the specifics, and they will inform you of the choices available. We always ask to speak to the chef.
For more information, including a list of specific counter services restaurants which are very accommodating to food allergies, please check out this webpage:
Special Dietary Requests.
If you need to take in your own medications, such as Epi-pens, antihistamines, etc... simply inform the Security Cast Member at bag check of the medical/allergy condition.
I hope this is helpful!