Yes dogs can be in the car and in parking lots. Just not on hiking trails.

Yellowstone you’d probably have to look around Bozeman or Jackson. Frankly I paid more for the $40 one because it was good for the dog. There was a cheaper one but he would just sit in the kennel.

Yes, I was just about to check those out. I hope it's more up to date than some other sites, but it's always a good way to start. thanks!

Hi do you happen to remember the name? I also found one, which is just sitting in the kennel which I don't want for her either.

The one I went to was camp bow wow in the Denver area. That might not help you as much. As there’s probably only one in Salt Lake City.

It would really help if you gave us a basic itinerary. Looks like SF via Yellowstone to Lake Lure in NC?
Might be better to stay north, depending like how long do you have for this road trip. Looks like nature is your main goal?
Lots of BLM and NF out west where your dog can run free.
Are you willing to get a tent and some gear for camping? Camping would be your best bet for the dog in NF-BLM. Many motels allow pets like Motel 6.
If you stay high in elevation is the southwest you should be fine and only go to places that allow dogs, at least on a leash so you don't have to leave him/her in a car. The nights will cool down nicely!
Some NF have cabins along with state parks, not sure about BLM.
For example, why Salt Lake City?
Go to Moab via Dinosaur NM and stay the night in Vernal.

Honestly, visiting National Parks with a dog simply isn't worth it, with the exception of drive-by sightseeing and the few NPs that have kenneling options .
This is pretty much the tall and short of it. There are kenneling options in Moab for doing Arches:
https://www.nps.gov/arch/planyourvisit/pets.htm
There are no kenneling options at Bandelier that I'm aware of, and if you are visiting the park during the hours you are required to take the mandatory shuttle, your dog will not be allowed on that so give it a pass.
https://www.nps.gov/band/planyourvisit/shuttle.htm
Areas where pets are allowed are very limited at Mesa Verde but there is kenneling at Morefield campground:
https://www.nps.gov/meve/planyourvisit/pets.htm