Call 01980 674763 for info on current restrictions (recorded message).
All of the tracks will be marked off with red flags if there's no access, but phoning beforehand may save a detour after coming across a flag unexpectedly.
Imber Range, at the Warminster side of the A360, is no public access at any time, other than the perimeter path.
On the main part of the Plain, the other side of the A360, there are a couple of tracks across the interior of the Plain, shown in green on the OS 1:25000 Explorer maps.
The main one, from Urchfont/Chirton down to the (ex) Bustard Inn, is a decent gravel road on which quite high speeds are possible, and is open most weekends (afaik). At the southern end there are the remains of the old tarmac road, which requires a certain amount of care to avoid sharp edges. Don't go through puddles at speed, as you don't know what may be lurking under the water.
It's largely downhill from north to south.