Can I purchase tickets on-line for the ferry from Woods Hole to Vineyard Haven ?

You cannot buy passenger tickets online yet.

Yes

If u are going by car. Otherwise u go to terminal @ boat and buy walk on tickets. Close to boat. Takes 2 minutes. Leave time-extra half hour- to park and take free shuttle bus to terminal in parking lot.

Yes and from some of the businesses close by

I had to ring them to buy tickets.

You can for cars, but I think you need to go to the terminal for passenger/bike tickets.