I tend to agree with TonyK regarding wild camping on the Ridgeway. It’s not really that wild. As to asking local landowners, you might be lucky but my experience of Oxfordshire landowners is not very positive. I’ve been chased by cows, walked across large fields with 20 bulls eyeing me up, had to turn back because rights of way have become impassable … When I was doing the Thames Path I even had an Oxfordshire stretch where the local farmer had sent his cows along the narrow towpath making it an appalling mud bath that made Kinder Scout look easy walking. And you’ve seen my comments on the off road usage of the Ridgeway – I don’t think it would be defamation of character to suggest that the local farmers are partly (though not necessarily wholly) responsible.
Not sure where you’re missing a place to stay, but there are quite a few villages and pubs not far off the path. You might be able to persuade a pub landlord to let you put a small tent up in his garden so long as you eat at his pub and have a couple of pints.
Also, don’t be tempted to stick up your tent anywhere near Chequers. Nice area, but you’d probably find a dozen armed police officers on your case. If you don’t know, the Ridgeway goes through the grounds of Chequers, and I always have the feeling I’m being watched when I’m using that bit of the path.
Your mention of tents? I had a bivi bag once and would strongly suggest you forget it. They’re OK if the weather is wonderful, but you don’t want to be in one during a summer thunderstorm (this is from experience). Also remember there’s nowhere to store your gear if it rains, and getting ready for bed if it is raining is not a lot of fun either. If you really want to go lightweight there’s a number of options here:
http://www.ultralightoutdoorgear.co.uk/For trekking I use the Golite Lair 2, which weighs a pound (450g plus a bit more for a groundsheet or use an opened up bin bag) and if you’re carrying walking poles you can use one of them to put it up (further reducing weight). There’s something nice and unclaustrophobic about an ‘open-ended’ tent. That, combined with a down 2 season sleeping bag means the camping aspect of a LDP doesn’t need to add excessive weight (or volume). As the Lair 2 is rated as 2-3 person, there’s room to hold a party as well as keep your gear dry.
Enjoy your walk.
S.