Two places come to mind, and what's more they run a bus that visits both camp sites, and they run late, so no need to take a car into Sidmouth. Try:-
Salcombe Regis Camping & Caravan Park
Salcombe Regis, Sidmouth EX10 0JH   01395 515303

or:

Kings Down Tail Caravan & Camping Park
Salcombe Regis, Sidmouth EX10 0PD   01297 680313

I have stayed at Kings Down Tail, and it has excellent showers etc. plus as I've already said the minibus service which is not at all expensive. Hope I have been of help