You can rent an old camper like mine (or one a decade older) from a couple of places around England - they are really
popular there right now. I had to look up Sprite - those look fantastic! I've seen a lot of that type restored around and about and I think you may be right, I do recall seeing someone with a small fleet of them, but I think that they were 'permanently' parked, at least the one's that I saw. You could rent them for the weekend but only to camp in on site, not to pull off for a trip.
The thing that I really want to rent is in Ireland - Gypsy Wagons they call them, but I don't want to hijack this thread so I'll stop there.
As others have stated, if you go to the trouble of restoring a vintage RV/Camper/TT/etc. (or keeping in good shape from the beginning) it would be a tough call to let it out to someone. I'm not sure how they keep those things going in England, but I've bent over backwards to keep mine in good shape over the 20 or so years that I've owned it, and I would never let anyone else even drive it