|After a few years break from annual NZ trips I went back this April and we hired a small car (from rent a dent in christchurch, excellent service). Probably free camped a third of the nights, but this is pretty tricky these days. Many of my old "secret" spots now have signs explicitly banning camping, and the number of tourists and campervans (and toilet debris) has risen dramatically, so I can't blame local areas for banning, and policing areas. Still possible, but usually means setting up camp after dark and packing up early. As John says, you need a certified self contained campervan with a toilet to be able to park up, making the cheaper vans not much of an advantage.
However with nzac membership you can get good value accomodation in the Darrans, Fox, Arthur's Pass and Mt Cook village. And many campgrounds have small cabins that are ok value if you don't take a tent. But booking well ahead, especially around major towns like Dunedin, Wanaka or queens town is advisable at that time of year.
Check out Mt Somers with its nearby DOC hut. I have yet to climb there, but it was a good walk one year when I had busted ribs.