The airport is Beef Island (EIS). Most airlines transit through Puerto Rico (SJU) and then connect to Beef Island. There is a new service offered by American Airlines with a direct flight from Miami (MIA) each day. Once you land in Beef Island you can pick up your rental car and drive to the villa which is about 45 mins away.
Alternatively you can fly into St Thomas, USVI (STT) and get a ferry over to Tortola. You can depart STT from the main ferry terminal in Charlotte Amalie or in Red Hook. There is a ferry that runs directly from Red Hook to West End, which is the closest ferry terminal on Tortola to the villa. The drive from the STT airport to Red Hook is about 45 mins. When booking your ferry tickets in advance, please note that it may be beneficial to take different companies depending on your flight times from STT. You should be at the airport 3 hours in advance so we recommend leaving Tortola 5 hours before your flight. We know this sounds excessive but if the ferry is delayed leaving Tortola, you will have plenty of time in STT before your flight.
Alternatively, there are private water taxis. If this is something you are interested in, we can help you find the best company.
Check http://www.bestofbvi.com/info/info_bviferry.htm for ferry schedules, you are looking for Charlotte Amalie, St Thomas to West End, Tortola (Or Road Town).