Most special diets can be accommodated at Walt Disney World. There is
an entire department at Disney Dining dedicated to helping guests with
medical and
religious-based food requirements. Just send an email describing
your situation, at least 14 days in advance of your trip, to
SpecialDiets@DisneyWorld.com.
A cast member will get back to you.
My sister has PKU, which involves severe protein restrictions. I understand the challenges of traveling with this type of food limitation. With any food issue, eating at table service restaurants is easier. When you make your reservation for a table service meal, they make a note on your file of your restrictions so that the staff can be prepared in advance. A chef will also come out to the dining room to speak with you to make sure they completely understand your needs. Disney does carry some low and no protein foods that can be made at table service venues with advance notice.
Eating counter service is, frankly, a more challenging. The counter service places have a book of the labels of the food served, but it is up to you to request the book and make your own determination before ordering. When eating at restaurants with my sister, we tend to stick with items that we know will be low-protein: fresh fruits and veggies, French Fries, ices, etc. To map your plan in advance, you should go to the website allears.net and search on the word "menu." You will find menus for almost every restaurant at WDW and find the ones that feature safe items. I suspect that you will need to supplement your child's food. It is 100% acceptable to bring food items into the parks.
I urge you to take a look at resources such as the pkunews.org and Michigan-pku.org which have articles about tips for Disney travel. Another resource is a book called PassPorter's Open Mouse for Walt
Disney World.