Most "American Slices" do contain dairy, as it is a USDA requirement for the classification of "pasteurized process cheese product" that there is at least some cheese in the product.

I have made some exceptional vegan cheesesteaks with marinated seitan and pumpkin puree + nutritional yeast "cheese" sauce. The key is the bread - if you can head to a local bakery for a proper fresh hoagie roll, the rest will follow. I personally like grilled onions and Bavarian mustard on mine. Fantastic sandwich - just don't put lettuce and tomato on yours!