Not sure about online list of B&B's, can only suggest that you get the Whitby tourist folk to email or fax you a list out....Unless you are happy to get a wee cottage, I know Whitby Holiday Cottages had a few left as they now have online booking so you can check availability, you can only book Sat to Sat online though but can still see if there are still some left, if so you can phone your weekend booking.
:)