We were at Yellowstone in July 2007. It was 76 degrees in the daytime, too hot to leave an animal in the car. We didn't have our dog with us, primarily because we were concerned about managing his needs. As I recall, you can't take them hiking on the trails, nor on the boardwalks. We saw animals in the campground and on the roads. I would not want to risk a dog vs. bison or bear encounter (we saw both, at close distance.) Most of the boardwalks are not fenced, so it would be very easy for a dog to take a flying leap into a thermal. Even worse, if a child or elderly person was to be suddenly scared or tripped by a dog *they* could take that flying leap into a thermal. Think it can't happen? Read "Death in Yellowstone".
Yes, you could take your dog to Yellowstone. I wouldn't.