Hello, Nica!
Not to worry! Walt Disney World does cater to Guests with dietary concerns, many of whom have Celiac Disease.
First, take a look at this webpage on Walt Disney World.com which details Park Policy regarding
Special Dietary Requests. You'll see that you can take your own food into the parks, however, they'd have to be previously prepared as those items cannot be prepared in a Walt Disney World restaurant.
Next, head over to allears.net, a fantastic unofficial site, where you'll find lots of information on traveling Walt Disney World with Celiac Disease. Simply click on "Dining" under the Walt Disney World tab and scroll down to "Special Dietary Needs." You'll see a link right there to "Celiac Sprue" which will then connect to blog postings, reviews, helpful hints and dining suggestions.
Finally, when you place your reservations as well as when you sit down to dine, make sure that your server is aware of your dietary concerns. At a Table Service restaurant, the chef will be happy to come out to speak with you personally about every option on the menu.
When we were in WDW just last month, one of my little nieces has a nut and shellfish allergy; without exception, the chef at every restaurant came out and discussed the menu options with her mom and made very clear what should not be consumed by my niece. At Kona Cafe, at Disney's Polynesian Resort, the chef even made it clear that he had a separate cooking area for her pancakes, since macadamia nuts would have been included in others' pancakes.
Have a wonderful trip! Be sure to send more questions as you think of them, too. I love to help!