We did this, but 7 years ago now (still my favourite trip/place ever though!). Went from Auckland to Queenstown in 3 weeks.
What time of year are you going? Can make a massive difference in terms of hours of daylight for driving and for road conditions. May also influence whether you want heating/AC in the vehicle.
Do you want to wild camp at any point or are you happy to stay in holiday parks? Will guide you as to what facilities you need in the vehicle.
Are you thinking of doing both islands? As a one way trip or a round trip - will need to think about when and where you are going to return the vehicle.
We used Discover NZ for our vehicle rental (a 2 berth, converted Mercedes sprinter). They were really friendly and easy to deal with, and happy to help if we had any issues during our trip.
General advice would be don’t try to fit too much into each day - there’s so much to see and do that it’s useful to have some flexibility. Likewise plan as many 2+ night stays in your itinerary as you can, as it gets tiring moving around all the time (but at least you don’t have the hassle of packing/unpacking).
Once you’re there, the iSites (tourist info centres) are fabulous for finding and booking places to stay/activities to do. But I would recommend at least having your first couple of nights booked so you don’t need to worry about finding somewhere. We then usually booked the next place whilst at each location so we knew where we heading.
Hope that helps. Happy to help with any further questions.