It depends upon the location - and I would certainly ask before accepting a position. I would expect it to be in a written description of duties and rules.
In general - locations in more wildlife popular locations will have more restrictions on pets. You usually don't want to be a camp host in National Forest campgrounds in bear country or mountain lion country with a dog.
I've seen several federal postings for camp hosts be very specific - no dogs.
Outside of a couple western states - most state parks seem to be dog friendly.
Another resource for information - look at the general camping rules for the park you are thinking about applying to host. What to the general rules say about dogs. If they are the normal rules - then dogs are likely okay for the hosts. If they seem restrictive - then the hosts might not be allowed dogs.
Also be aware that as camp hosts one of your duties is to make sure folks clean up after their dogs.