Pickering, around 30-35 miles NE of York with Sunday shooting on its own (300 yard) range.
http://prandpc.org/
The club is a one of a group of regional clubs which has bookings made by the Yorkshire Rifle Association on theior behalf for Strensall MoD ranges which are 7 miles from the city centre to the north and which host 300/500/600 club shoots on gallery type targets. There should be two weekend bookings a month for Strensall, but this is rarely achieved because there is often no military use on the day, the few or often sole cadet or Terry booking being cancelled usually only days beforehand.
Club secretary is Don Garland, tel 01287 651049
There is also The Railway Institute RC which shoots indoors 22 on railway property close to the station and has a rimfire / pistol calibre outdoor range at Hessay near the York-Harrogate railway line a few miles NW of the city.
To the south, there is Hensall RC just down the road from Eggborough power station and bordering the M62 maybe 8 or 9 miles south of Selby, so around 25-30 miles south of York. It uses old sand quarries for rimfire and pistol calibre rifle shooting and BP pistol, three covered FP ranges for 25/50/100 metres.
Some York people I know are members of Mattersey RC on the South Yorks / north Notts border area.
http://www.matterseyrapc.org.uk/
Then there are some ranges / clubs to the West on the M62 corridor:
Paul Lane RC as was (100m outdoor + indoor) which now seems to be a commercial range venture called Mirfield Rifle Range, about 5 miles south of the M62 J25.
Lydgate RC which uses Deer Hill Range near Marsden, a 500 yards private range. I used to compete in HBSA 'Open' historic arms comps there - nice spot if you like wild uplands areas but can see abysmal weather (even worse than Diggle). Does have a nice warm range house though.
Finally, on outdoor ranges, PSSA with a 1,000 yard range complex at Diggle just inside Lancs. Outdoor ranges at every distance from 100 to 1,000. Almost every centrefire discipline from Benchrest to Practical Shotgun / Rifle / CSR covered with competitions every Saturday and Sunday year round. Covered benchrest equipped firing points at 100 / 300 / 600 / 1,000 as well as open grass FPs at 200-1,000 yards. Heated range house with kitchen facilities. Subject to RO availability and ranges not hired out to other clubs or organisations / companies (BAe Radway Green does ammo testing a couple of days per month there for instance), the range is open to members on weekdays too.
The Gun Pimp on this forum is the membership secretary. From where I live just south of York, it's 60 miles door to door and 1 hour 15 to 1 hour 30 mins being mostly dual carriageway and motorway.