This fast casual eatery can be found throughout the Southwest (California, Nevada, Arizona, and Colorado at the time of writing). In their online nutrition information they actually have a Guests with Food Allergies section. Their food allergy lists can be a bit confusing; if you click on dairy, it will give you a menu showing all items that contain dairy (rather than the ones that don't, which might be a bit easier, since you don't really have a full menu to look at it side by side). However, they do break down what component of the meal has dairy, which is quite nice. For example, most of the burgers without cheese are actually dairy-free.
I visited an Islands recently and their food is good. Nothing overly-exciting, but good. I enjoyed their Toucan Chicken Sandwich sans cheese. Plus, the fries are made without dairy (double check that this hasn't changed though!), and are now included with burger and sandwich orders. – They didn't used to be.
Main Islands website with location: www.islandsrestaurants.com