Been there twice in campervans, once with Wilderness who have now moved to the high end and this year with Kiwi Campervans, both times in a VW T5 hightop.
Not a cheap way to do it, budget on about £180 a day all in including your van, insurance, site fees, fuel, food etc but excluding flights.
As advised above, I would stick to doing one half of one island in 17 days otherwise it will feel like a military operation, like you are just on the road all the time. The roads are good but much like Scottish A roads you won't be getting anywhere fast. 250-300km feels like a long day's driving in a 'van.
We never had any bother finding sites, it's best to phone ahead in the morning if you're heading to the tourist spots though.
I can't recommend the Rankers NZ App
https://www.rankers.co.nz/about/official-camping-nz-app highly enough as you can download the maps to your device for offline use as phone signal / wifi can be patchy.
At the lower end of the market you will be pretty much 'on your own' once you've left the depot, a quick look at the reviews on Rankers will tell you that.
Free maps are available from NZ Open GPS Project and worked fine in our Garmin. I've got a load of gpx files and other stuff, if you're interested PM me.
Have fun, we absolutely love the place and intend to return once circumstances allow.