I don't think they do? We stayed at a site this weekend and the App wasn't even mentioned at check-in. I've looked at their website FAQ and it says:
https://www.caravanclub.co.uk/coronavirus-faqs/
The current Track and Trace system is in use at all Club campsites and utilises our reservation system in order to minimise the administrative burden on our members. On 24 September, the UK Government introduced a new NHS QR code system within England and Wales in addition to the current Track and Trace procedure. A similar system was introduced in Scotland and Northern Ireland. This new system allows individuals to download a Government App and then scan a code within hospitality venues, to “check in” to the venue - this is entirely voluntary. Whilst not a requirement for members to “check in”, the Club, and other campsites or holiday parks, are legally bound to display the new NHS QR Code posters at its campsites.
We feel strongly that containing outbreaks at an early stage and reducing the ability for the COVID virus to spread is ultimately beneficial to us all and are asking our members and guests to support the use of the new system.
The existing Track and Trace system that is already in place will continue, should members not have the ability to download the App.