I’m struggling to think of anywhere legit that’s nearby, because so much is World Heritage, National Trust and English Heritage.
There are places along the Ridgway where you might be able to sneak a crafty bivvy in one of the beech hangers just off the path, one or two are quite dense and don’t get too many people in them, these are between Hackpen Hill and Avebury itself, but it’s all farmed up there, and there’s some woods with shooting rights close to Fyfield Down, alongside the Sustrans trail that goes across the Ridgway towards Marlborough, where you might be able to find a well-covered spot that’s fairly well hidden.
There’s some fairly thick gorse with open areas down in the valley that you could get into and set up a bivvi with a cover, but it’s all quite open, and it would need to be a late to bed, early to rise to avoid any issues.
I would avoid the top of Silbury Hill, it’s a bit draughty up there…
The Wagon and Horses at Beckhampton have a garden area out the back, it might be worth a cheeky phone call to see if they’d allow an overnight pitch, you never know.