What type of facilities are you looking for? There is a yard at Alanbrooke barracks just outside thirsk towards topcliffe, they have a school and walker, there is a bigger yard in Northallerton with indoor and outdoor school but is very busy with BSJA comps, there is a yard towards felixkirk, not sure what, if any facilities. I am sure there are small yards in thirsk with just stables and turnout as i have seen horses in fields but i dont know any of these yards. If you want to go further afield there are yards towards and in Ripon too. (oh just remembered, there is a farm just outside thirsk towards ripon (by the small bridge by the turning to the A1) that is being converted into a yard, there is a massive barn type thing that looks like it will be an indoor school)
There is Richmond equestrian centre if you are looking for better facilities if you dont mind travelling that far. Hope that helps!