Try www.southerncountiesleisure.com Not in Wilstshire between Yeovil and Dorchester but it is a fantastic place full of civvies who enjoy shooting. Also a go-kart track. I sometimes hold team building days there and they are really helpful and good fun.
There's no need to worry about mixing with Green-jobs as Larkhill Clay Target Shooting Club is run as a civvy club, but is located on the RSA at Larkhill. It's open 2nd & 4th Sunday of the month. E-mail Keith Howe at email@example.com or 07751 167883 for details. It's a good ground with decent targets, but very flat and open. There's a scoff hut on site, no restriction on plastic wads and is generally quite cheap: Â£20 for 100 birds sporting, IIRC.
A bit further north, on the A345 is The Shooting Club at Upavon. Another nice ground with a decent club house, good targets and use of topography - open 6 days a week (closed on Tuesday). Â£25/100 sporting and (I believe) no restriction on plastic wads.
Alternatively, Laines Shooting School just off the A303 near Andover is open weekly Mon - Sat (although it's closed on Sat 21st Feb for a charity event). Nice ground, good facilities but costs more per round and you have to use their (low noise) ammo due to locally enforced noise restrictions. Last time I shot there was 2007 and I believe the cost of 100 birds plus their ammo was around Â£45.