Looking for a lovely pub with accommodation or a B&B within short walking distance of a pub for a trip to Bempton Cliffs
There's only one pub in Bempton itself, the White Horse. I've not been in for a few years but I didn't notice it having accommodation, though I hear the food's good now. You may have more luck looking for a pub with rooms in neighbouring Flamborough.
It's a bit of a trek up to the cliffs from Bempton village so you'd be likely to be using a car to get up to them anyway.