Archery appears to have come to New Zealand with the advent of the Europeans. An account of the first interaction by Captain Cook is here: Victoria University. The Tahitians were very skilled with the bow, yet the Maori had little concept of it. Here is another article which also references the words that Maori used which had references to bows, yet they did not use it themselves.
The earliest settlers from Europe started their own archery societies, as referenced here by Murray Cramp: Colonial NZ Fetes.
Not long afterwards, the first Auckland Archery Club was born – see this article by Murray Cramp: Indoor Archery History.
From the late 1890’s to the early 1940’s, archery appears to have been a minor passtime in New Zealand, with few organised clubs. It was after the war that many soldiers started their own clubs, starting with Dunedin Archery Club, and Auckland Archery Club, Randwick in Wellington, and Gisborne. Many others followed.
Since then Archery in New Zealand has become an established sport, with many clubs in Target Archery, Field Archery and Bowhunting. Archers have gone to the Olympics and Commonwealth Games (where Neroli Fairhall won a gold medal), represented and won medals at the world Championships, and even held world records.