Pakatoa Island is a small privately developed island in New Zealand's Hauraki Gulf, and the honest starting point is this: there is no car rental available on Pakatoa itself. It is a tiny island reached by boat or private launch from the Auckland region, built around a resort and residential development rather than any kind of transport hub, and its size makes a car largely unnecessary once you are there, since walking covers the whole place easily. Anyone searching for car rental tied to Pakatoa is better served thinking of it as a boat-access getaway rather than a driving destination, since no operator maintains a fleet on the island itself.
For travelers who do want a rental car as part of a wider Hauraki Gulf trip, the practical move is to arrange it on the Auckland mainland before or after visiting Pakatoa, since Auckland Airport and the city center both host the full range of international and local rental companies. From central Auckland, the waterfront and ferry terminals used for Hauraki Gulf islands are only a short drive away, making it easy to combine a mainland rental car with a separate boat trip out to Pakatoa or nearby islands like Waiheke and Rotoroa.
Driving is on the left throughout New Zealand, matching the same side as Australia, South Africa and the UK, and Auckland's roads range from modern motorways to winding coastal routes once you head out toward the gulf-facing suburbs and ferry docks. Traffic around the city can be heavy during weekday commuting hours, particularly near the downtown ferry terminals in the late afternoon, so building in extra time before a ferry departure is sensible. Since Pakatoa itself remains car-free, the simplest approach is to treat any mainland Auckland rental as bookending the boat journey rather than extending onto the island.