taking a cat caravanning is fun and a great conversation starter. We did it for many years and still miss our mog but it's not possible now.
Cats really do need to be on a lead on sites - but they have relatively weak necks so need a body harness like a small dog. Ours travelled in a medium sized pet carrier - the plastic type which comes apart along the centre line - which also served as her bed home and away so she was very much at home with it. In transit, put it on the back seat of the car and secure with the central seatbelt. Face it forwards not sideways so she can see you and out.
Our vans have had sepeerate showers, so I made a false floor from plywood and lino covering and on this went foold dishes and litter tray. A door stop prevented the magnetic strip completely closing the door so cat had access at all times.
On site we used the screw into the ground type dog tethering post - cats will tangle up any known tether so make sure they are out of range of van steadies which are normally greasy. Where they sleep at night is up to you, but do not be surprised to find them way down under the duvet.
If you take them abroad you must, of course, follow the rules for pet passports and checks but pet food is increasingly available in supermarkets everywhere - often the same brands.
Ours was too old to be chipped and travel when we started wintering in Spain, but after one trip I paid more to the cattery than I had in site fees, so there is an economic case as well.