The NSA is definitely the place to look, they're really nice if you send them an email and will advise of local farmers.
You don't have to stay with the family if you can drive or get a lift there and it really depends on the system. The first farm I went do I did 8am-5ish, the farmer would check them at 10pm then leave them until 6am but when I could drive and moved to Uni we did the 8pm-8am night shift on a bigger Organic farm so we were busy all night pulling lambs.
I wouldn't worry about being a veggie too much, most farmers are meat and two veg kinda people but if you're not fussy otherwise it shouldn't be a problem. A few friends have stayed on farms and just had what the family have eaten minus the meat.