I wondered this about both La Hacienda de San Angel and La Cantina de San Angel. I am gluten and dairy free. Here's what I got back from the assistant to the director of operations at the restaurant.
"Per the Chef, Everything that they serve is Gluten-Free, except for the
following items:
In La Cantina:
-Chicken Tenders
-Melted nacho Cheese, that are used for the nachos
-Churros
In La Hacienda:
-all Desserts (except for the ice cream)
-Shrimp Tacos
-Some items come with Flour tortilla, but that can be substituted with
corn."
I ate at La Cantina for lunch and the tacos were fabulous. So good, I'm going back in the next few days when we head down to WDW. It is important to note that the salsas at La Cantina that were available on the fast food counter (with napkins, silverware, etc) was NOT gluten free, but the pico de gallo was. I'm sure you will have no problem at La Hacienda. Enjoy and hope that helps!