One place that you sometimes see kitesurfers is Long Bay, Beef Island. Empty beach (except for four days a year when half of Tortola is there "for a lime"), stronger winds because of the squeeze between Beef/Tortola and Scrub/Camanoe/Guana, and a fair sized broad reach to Little Camanoe. So back and forth, back and forth, between the two.

You could do it as a lunch stop, or dinghy over from Lee Bay, Marina Cay or Trellis Bay. In June you can usually safely overnight, no swell likely, although you rarely see a boat in there other than the Sail Caribbean kids.

Come to think of it, one of the four days half of Tortola goes there is at the end of June/begining of July - Territory Day, when they have the Fisherman's Jamboree there.