Couple words of warning here. Burritos are almost always going to be made with flour tortillas (wheat flour). Even if you can find a place that offers GF burritos there will likely be cross contamination. Also ethnic restaurants in general are not usually good at gluten free preparation or knowing everything that is actually in their ingredients.

My advice is find good GF wraps / tortillas and make your own burritos. Get used to doing this sort of thing (making things you never bothered making before) to avoid being 'glutened'.

Source: Am spouse of someone with celiac.