I've gone camping with pets a few times. I did edit the Granite Falls lots to have pet feeding automats (since single bowls might go missing), but @kalaksed told me that the ranger's station that sells tents and other camping equipment now sells food bowls, litter boxes, pet toys and so forth.
I've noticed that the pets behave somewhat normally on the 'home' rental camping lot. Eat from the bowl, go to the litterbox if feline, run around, hop on counters, but when you travel with them to other lots (not necessarily even in camping, I've seen this behavior in other normal non-home lots too) they sometimes just stand where they loaded in instead of exploring the surroundings. The sims can of course call them over, but they do seem to be a bit lost if not at home-lot.
Going to the National Park and hermit's hut especially can be a bit tricky with pets. To make it more annoying is that even if you leave your pets in your camping home lot, when you have another sim join you in the hermit's hut, the pets auto-appear there, and then just stick to that spot until you leave. Aka they didn't come with the sims to the National Park, but when one sim travels to hermit's hut, then has another sim join them there, the pets magically appear there.
So going to Granite Falls works fine with pets if you get them the stuff they need at your home campsite, but going around to other Granite Falls lots can be a bit tricky sometimes and need a lot of sim-guidance for the pets.