There are no camp sites actually in the Gorge itself but these are very close by:
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Lillypool Campsite and Cafe – Shipham Road, Cheddar, BS25 1RQ. Telephone 01934 741244.
Bucklegrove Caravan and Camping Site – 01749 870261
Rodney Stoke, Cheddar, BS27 3UZ
Mendip Heights, Townsend, Priddy, Somerset BA5 3BP.
Naturist camp sites
None in this area but this list on UK Campsite includes locations elsewhere in the South West.