My son does not have Celiac but is allergic to barley, wheat/gluten, eggs and peanuts (not airborne), with barley being the most severe. He has no problem with oats and, thank goodness, has no problems with dairy. We have taken him to WDW three times in the past 2-1/2 years and have never had a problem with finding him something to eat. The restaurants at WDW have bent over backwards for him, sometimes giving him more food and options than he could ever possibly eat. He has had special meals prepared at 'Ohana (they use Tamari sauce instead of soy), gluten free pasta and soy and egg-free dishes at the Crystal Palace and 1900 Park Fare. His best meals so far were at Boma (meat and veggie plate and egg-free ice cream for dessert) and the Hoop Dee Doo Revue, where they prepared a special chicken and veggie plate with a separate fruit plate that was big enough for our whole family.
When you make your ressies through WDW-DINE, alert them to your son;s condition and they will give you the direct numbers to the restaurants for you to call and speak to each chef. They will recommend that you call 7 days prior to your trip.
Have fun!