We camp with our mini long-haired doxie and 2 Cavalier King Charles Spaniels. They are one of the reasons we have a Class C (I work for a hotel company and could stay cheaper at our hotels
). We have never been turned away (usually camp in private camp grounds up and down the East Coast). As has been pointed out, in a few places (including Fort Wilderness at Disney) there is a small charge for having the dogs (not per dog) -- 5$ a night seems to be the average. While I don't necessarily agree (hey, they don't charge for kids), it is a small price to pay to bring them along. We have a fencing unit that we use to fence in our outdoor rug and it works well for the little guys. They are usually so tired after a day of play, they sack out early and aren't really barkers anyway.
I want to second the recommendation for Four Paws Kingdom Campground in Rutherfordton, NC -- it is a dog focused campground with wonderful activities for the dogs and adults. However, younger children aren't accepted and there are no activities for kids.