"Silver Apple" (Holland one tonner 1974) used to be on skippered charter from Amble. The touristy boat trip to Inner Farne runs, I think, from Seahouses. Obviously you don't need a boat to visit Lindisfarne but the whole atmosphere changes when the tide rises and the tourists leave.
You could try Paul at Bob Moncur Sailing (http://www.bmsailing.co.uk/). He is based in Hartlepool and does trip up and down the East coast, including the Farnes.