Hello Chris,
Unlike cars where it is a legal requirement for the car V5 to be registered with the keeper, there is no similar requirement for caravans. The CRIS scheme is a voluntary scheme, and it is useful, but its not a legal requirement.
The vast majority of storage companies are independent businesses, and becasue there is no standardisation of store terms and conditions that they have to follow, each company will have their own sets of T&C's and policies that dictate how they operate. I can only suggest you discuss the situation with the storage company, as only they can decide if you can use the site or not.
I can only assume the storage company may have had some problems where the ownership of a caravan has been in doubt, so they now ask for the CRIS document.