Going up that way soon and looking for nice sites to stay at with a bit of character (short walk from a pub ideally). 4 nights so 2 near the coast and 2 maybe more inland. Been recommended some elsewhere but some are just rows and rows of caravans and I know you are a more discerning lot 😉
Good luck is all I'll say. We've just been up there for a biking weekend and campsites near a pub were surprisingly thin on the ground, we ended up hiring a cottage.
Interested if you do find somewhere though...
Hutton le Hole?
Middlewood Farm or Hooks House Farm near Robins Hoods Bay. I've stopped at Middlewood which is a short walk to quite a few pubs.
Dalby Forest Campsite is at the entry to Dalby Forest and near the mtb trails, but no pubs in walking distance.
https://fletchers-farm.co.uk/
Literally at the bottom of the road up to gribdale (captain cooks monument), Roseberry topping, Guisborough woods (lots of bike trails).
Pub/licensed restaurant on site (don't know what it's hours are though), 10 minutes walk into Great Ayton with a couple of pubs, restaurants, petchs pie shop (definitely worth a visit), nrg cycles should you need a spare, short drive to stokesley.
There's a lovely site just outside Wombleton. Great facilities and 10mins down a quiet lane to a properly good pub.
Near Whitby, stayed there a couple of weeks ago, great laid back site. 5 mins driove to the coast. Google tells me there's a pub in the village too. We didn't visit it so can't confirm whether or not it was open or any good.
Lord Stones Country Park* has a small campsite. I can't think of many campsites in the UK with more character. It's right on top of the hills and surrounded by trails. It has a licensed café / restaurant onsite but hours can bee odd. There are 2 pubs near-ish, including the Blackwell Ox which does good Thai food. It's on the NW side of the moors so very much inland.
*It's not really a traditional country park, it's a commercial venture but the location is spectacular. You can take a 5 min walk from your tent to one of the highest points on the moors.
Beacon Farm, Sneaton, near Whitby should be high on the list. Clean toilets & showers, view of Whitby Abbey and the sea, not too expensive, the family who run the farm (several generations) are friendly, park for the kids but quiet and none of the park within the white line holiday park rules.
There is a Bridleway from the farm that runs down to Ruswarp and then it's a short drag up the lane to the Whitby-Scarborough cinder track cycle route. I think that there is a pub a few hundred metres away as lots of people roll back into the campsite on foot (not taxis) late on an evening.
Wet weather options from there include all the tourist tat you never wanted in Whitby.
Stayed there on Saturday in the van and everyone had a lovely chilled time.
Oh, and it has an en suite ice cream parlour (ice cream made on the farm).
If was further from home I would book in for a full week.
Did I mention the ice cream!