As someone who has been severely gluten intolerant since I was born, I can say that this one thing that Disney has always done really well. Even before many restaurant/supermarkets were stocking gluten free items, Disney was one of the first hospitality brands to have these options. I enjoyed my first hot dog on a bun at Casey's Corner (at age 18 I want to say) and I plan on enjoying my first beignet next time I go down (at POFQ.)
Most food courts/counter service restaurants will have some options, such as a gluten free brownie, gluten free bun alternatives for hamburgers/hot dogs, etc. They will also use a separate frier for anything you get, so there's no need to worry about cross-contamination.
At a table service, they will typically bring out the chef and s/he will walk you through all the options on the menu, and most of them will have things that are not on the menu that you can order as well.
Be prepared to wait an extra 10ish minutes to speak to a manager about your needs, and for your food to be cooked special. Also, Babycakes bakery at Downtown Disney is all gluten free (and very tasty.) They also have the new Garden Kiosk at Animal Kingdom that stocks some gluten free products as well. There are some restaurants that are better than others and will have more options (Sanaa for a table service and Sunshine Seasons for a counter service are the two that come to mind.) But really, you can't go wrong pretty much anywhere - they will have options for your wife.
I'd just steer clear of the pizza - I've never enjoyed their gluten free pizza. In fact, the ONE TIME I enjoyed a pizza on Disney property, I thought "oh my god, this is amazing!" and I thought that they had finally found a supplier with better gluten free pizza. A few hours later, I had to leave my seat at Fantasmic because I was suddenly very sick. Needless to say, they served me a gluten pizza when I was very clear that I had severe allergies. I went to City Hall the next day and they wouldn't even give me a refund on the pizza, let alone anything else. After I got home I called guest services and explained that not only was I really upset that they gave me the wrong pizza, but that they wouldn't even refund me for making me very sick - they usually take food allergies really seriously, but this was a big slip-up. Be very clear that your request is for health reasons, not just because gluten free is suddenly a hot trend. (Some restaurants, not at Disney, claim things are "gluten free" because they are playing into the fad, and are not looking out for people with actual reasons for needing to eat gluten free.)
Enjoy!!!