There used to be some daysailers out of Village Cay, Tortola. Maybe still are.
Also, it might be possible to charter a boat with a captain for a few days from one of the charter operations: Moorings, SunSail, etc.
It would be worth an email to them asking what's available. Also, try the BVI tourist board and advertisers:
Official Website of the BVI Tourist Board – British Virgin Islands
From East End, you can take a boat to the Bitter End, Virgin Gorda, where they have a variety of day sailers and expeditions.
Bill