Welcome to UKCS. There’ll be dozens if not hundreds of sites, you need to say where you’re interested in. I could list very many (if I had time) but they’re independent sites not CLs or CSs.
If you use the Site Search, you can limit it to CLs & CSs and tick on “dog walk from site”. But it’s not a perfect tool. I’ve found it can seem to apply only if it’s in the site ownership or control (a wood beside the field, for example). Conversely, several sites had a dog walk listed (hence our choosing them) but the walk was along a busy road verge. I know dozens of sites where we can walk straight from our pitch into off-lead areas, but they’re not listed as such in the directory.
Still no reply. We are at a CL now, Carefree Nurseries at Titchfield which has a dog exercise area. I chose this site because the reviews state that you can see the shipping in the Solent. You can't, the hedge is too high which is disappointing, but that's another story.