Many private RV campgrounds that we have visited have pet restrictions, either by choice or because of insurance requirements. Most common are a 2 dog limit and/or weight limit of 20 pounds. Many also restrict breeds. I have been to parks with limits that have allowed guests with show dogs, often exceeding the weight or number limit. It would be wise to travel with a camping directory, like the Good Sam, Trailer Life, or some other book. Also cross reference with RVparkreviews.com and with the campground website. And always call ahead and make your reservations with full disclosure.
Happy Trails.