Parking anywhere in Avebury will be difficult if not impossible. Having said that you might still be able to get a B&B in the area and, if you can, you should get a parking place to go with it; you'd then have to walk or take a bus into Avebury itself. Failing that you could park outside Avebury but you'd still be looking at a 20 or 30 minute walk to the circle/village itself (and you should be aware that the area is notorious for car break-ins at all times of the year).

Devizes, Marlborough and Swindon all have good and affordable accommodation - you might want to consider booking into somewhere there and then bussing or taxiing into Avebury - not ideal but a lot less hassle.