permitted to leave caravan unattended?

Hi sue, I do not know if a CL in the area, but you might try asking at this small commercial site.

https://www.exeterviewstables.uk/

https://www.ukcampsite.co.uk/sites

If you access above website and put Exeter into search for a site, then put storage into filter you will get about 12 sites that offer storage. You will have to contact each to see if any of them willing to offer what you are looking for.

Sorry can't help from personal experience.

Best check your caravan insurance too, some specify occupancy when onsite.

Wonder if OP found out responses to their liking, or perhaps not, as they have not bothered to come back and say one way or another?