Hi Julie!
As the parent, you'll have to determine the best course of action but I can reassure you that WDW has an excellent
Special Diet program which includes accommodating allergies.
In the theme parks, there are at least two counter-service (quick-service) locations that can accommodate guests with food allergies and other special diets. Follow the above link to see a list of those locations.
Trying to dine at table service restaurants for all of your meals will definitely take a lot of time away from touring or just relaxing at your resort. Can you have breakfast in your resort room or at the hotel's quick service location? Feel free to ask to see the Nutritional Information Book at the quick service location to insure that any items ordered safe for your daughter!
I would also visit www.disneyfoodblog.com. This site has excellent Special Diets resource pages. Under the Dining FAQ page, you'll find the link!
I wish you a safe and magical vacation!!