I tend to agree with TonyK regarding wild camping on the Ridgeway. Itís not really that wild. As to asking local landowners, you might be lucky but my experience of Oxfordshire landowners is not very positive. Iíve 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 Ö 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 youíve seen my comments on the off road usage of the Ridgeway Ė I donít think it would be defamation of character to suggest that the local farmers are partly (though not necessarily wholly) responsible.
Not sure where youíre 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, donít be tempted to stick up your tent anywhere near Chequers. Nice area, but youíd probably find a dozen armed police officers on your case. If you donít know, the Ridgeway goes through the grounds of Chequers, and I always have the feeling Iím being watched when Iím using that bit of the path.
Your mention of tents? I had a bivi bag once and would strongly suggest you forget it. Theyíre OK if the weather is wonderful, but you donít want to be in one during a summer thunderstorm (this is from experience). Also remember thereís 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 thereís 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 youíre carrying walking poles you can use one of them to put it up (further reducing weight). Thereís something nice and unclaustrophobic about an Ďopen-endedí tent. That, combined with a down 2 season sleeping bag means the camping aspect of a LDP doesnít need to add excessive weight (or volume). As the Lair 2 is rated as 2-3 person, thereís room to hold a party as well as keep your gear dry.
Enjoy your walk.