I don't know of any sites within walking distance but there was one I looked at last year at Colonley, which was close to the railway station and just one station along the line from Skipton.
I'll have a look to see if I can find the name - it was either a CS or CL with one of the clubs - I'm sure someone else may come up with the name.
Skipton - Riverside Caravan Site
Moorfoot Lane
Cononley
Nr Skipton
North Yorkshire
BD20 8LX