We used to have a caravan at Western Sun Folk, that also had a winter storage area. The annual membership fee worked out cheaper than off site storage for the winter months. Site also had good security, so was easy to insure.
Depends on how far from home you want it to be.

Tything Barn.

Acorns near Tiverton Devon do caravan storage may be too far away from you?

St Anne's Cottage on the edge of the new Forrest.

Take a look at Spielplatz, not too far from you and they do winter storage as well.

The Naturist Foundation has pitches and winter storage, A really well run club, very active, but not cheap..... but you get what you pay for