I was there in June - not camping, but did check out the camp sites to see which would be best for a future trip.
Troytown on St Agnes has fantastic views out to the Bishop Rock Lighthouse, but very exposed. There were several bell tents - you can hire one from the campsite.
St Martins is right down by the beach. Not my favourite island (I get fed up with the long road from Higher Town to the other side). Has Adams fish and chip restaurant for eveing meals. The Sevenstones pub was still open when I was there (very run down). It's now in receivership but hopefully it'll be back in business next year.
Bryher campsite is in the middle of the island. Very peaceful. I think this would be my first choice. (Love Popplestone beach).
St Mary's campsite is up on the Garrison so quite a hill to climb to get in and out of Hugh Town. Great views from the top.
I adore the Scillies - there is so much to do. During the week we went to 2 evening quizes, 2 lectures, sailed out to the Bishop's Rock Lighthouse, circumnavigated Bryher, St Agnes and St Martins (legs are still aching from all the walking!).
Next year it I shall find a quiet beach and leave the wine chilling in the sea.