Another possibility that might work out is a travel trailer and a full-size van. Many of the full-size vans are (or can be set up to be) quite substantial tow rigs, and have plenty of room inside for dog crates etc. Obviously pulling a fifth wheel would be impossible.
As MetalGator wrote, you will have to plan ahead to find places to camp where that many dogs are permitted. Do be aware that not infrequently public parks with campgrounds may allow pets in the campground but have stricter restrictions in some or most of the rest of the park.
AAA does offer campground guides (I think it's Woodall's guides now) free for the asking to members. Most here use various online websites, such as
rvparkreviews.com or google maps searches or a few others, or smartphone apps (RVParky seems to be a popular one). Generally you'd have to look at the web site for the particular park, or call them on the phone, to get specifics about things like how many dogs may be permitted and what extra fees, if any, would apply.
