I think this would be a great business opportunity for someone. Definitely, it's a new open market and a service I'm sure lots of people would use. The problem would be to connect your availability to pet sit with a comparable campground / RV park that wouldn't take the idea and give the job to relatives!
For those on the other end, (the consumer), what a great service to have available for your use. I know we would probably use it. Our dog owns us and our lives revolve around everything the dog can do or where he can go. Like over-demanding children ... sometimes a break would be nice!