Create an account on Airbnb and start searching, and also look at Vrbo.com .
You need to do a little more research ahead of time, since Airbnb/Vrbo rentals aren't standardized in the way that established RV Parks are. Sometimes you share bathrooms, or kitchen space, with the owner or other renter. Other times you will receive a lockbox code by email, and you arrive, stay, depart, and never even see another person for the entire trip.
Since each one is managed by an independent owner, the pet policies are highly variable. I've used both of these websites to find rentals that allow multiple large breed dogs.... but you do need to be flexible and open minded, since you're usually staying in someone else's home, and are subject to whatever rules they decide to enforce (or not).