Most of the Lakes YHA’s are ‘mtb friendly’, imo the best one is Hawkshead.
There’s a dedicated camping area at the side with pods & a round cabin type thing, there’s also a decent amount of parking. You’re right next to some brilliant trails in Grizedale & Hawkshead is about 1 mile away where most of the pubs do good food.
When are you planning on going? The problem you may have is that a fair few yha’s are closed over winter.
Also check with each individual yha what their bikestore is like- bring a good lock. Ambleside yha bikestore is just a garden shed..
Better to post some more details- preferred area’s, size/age of group, etc..