Yip Turkeys every where, on the side of the roads, farmers paddocks. bloody things.
South island there is Thar, Chamois, deer, wallaby, pigs, and the good old Canada geese. North Island has Deer, both reds and sika, Canada geese, wild pigs, and there is a two week special season on parry's. so plenty to hunt at that time of the year. i'm sure I've missed some but we have loads of game to have a crack at. You will want to plan ahead for bigger game though.
Just because you haven't done something, doesn't mean its not worth trying.