Many moons ago we traveled with our own Rice milk. I don't recall any dairy-free shakes or ice cream in the parks. If you're doing TS meals, let them know in advance and you might get some treats for him.
I'm not sure if you're dealing with a dairy allergy or lactose intolerance, but be sure to ask for the ingredients on things like Dole Whips. I have found many a "dairy free" smoothie to contain traces of dairy protein. This is a problem for those with true dairy allergies.