Disney is VERY good about gluten free. I have Celiac and have only had one issue at Disneyland (in several trips) and no issues at WDW. Just be sure to let your server know. They will usually have the chef or cook (QS) take your order and then bring it out front. They are very knowledgeable about cross contamination and which foods are gluten free. There is something gluten free (or that can be prepared gluten free) at nearly every location.
Disney takes food allergies very serious. My daughter has Celiac also. She loves our Disney trips because there are more GF options available to her than anywhere else....and she never worries about cross contamination because they are so careful.