I'm not 100% sure but think you might have to stay down the road near to Flamborough.
If you type the postcode YO15 1AU into Google Maps it will give you a good view of the area and a large camp/caravan site called Thornwick Bay -
Thornwick Bay & Sea Farm Holiday Centre - Home & Latest News Flamborough Holidays
You can also see Bempton Cliffs a bit further up so its not too far away. You'll probably (again I'm guessing) be able to walk along the cliff edge to get there.