Up tp a certain point yes they will. Two years ago we booked a water taxi for 4 people. The other couple had a flight delay such that they did not arrive until 7:30 PM. The water taxi company told us that we had to be in West End by a certain time (7 or so?) and would not wait until the other couple arrived citing customs being closed and not available for overtime. So two of us had a pretty expensive ride to Tortola and the other two spent the night in a hotel for a lot less than the cost of the water taxi.

Maybe if they/we paid extra to go to the airport or Roadtown (if it was open) they could have cleared there.