As per the vouchers the Airpoints partners are Avis and Budget (who are one and the same company). As per the Air NZ website (which is what is also on the back of the voucher)
Rental car voucher: Each year Airpoints Elite members receive a voucher for a complimentary, one car type rental car upgrade at participating locations in New Zealand and Australia. Rental car upgrade requests should be made at the time of booking, and will be assessed by the car company upon collection. They can be redeemed on a Friday, Saturday or Sunday and are valid for up to three consecutive days. Upgrades are subject to availability and some car types may not be eligible for upgrade. Vouchers cannot be used in conjunction with any other offer. Rental car partners may change from time to time.
You don't book the upgrade online, it's simply done at the counter.
I also never use Avis or Budget in NZ as Thrifty are soooooo much cheaper and with their MCAR deals and AA discounts you don't pay airport fees. The only way to avoid these with Avis/Budget is to have a good corporate account.