Whitby is horrible for people with mobility problems. Venues are very scattered, there are hilly sections and the buses don't generally go near to the venues. Your best bet is to find venues that run events for most of the day and just stay near to where those events are. I think the Rifle Club, for instance, runs events for most of the day and also has a small snack bar. You do have to climb up the stairs though.

Beg for a lift from someone or book the occasional taxi.