We arrived in Wellington a year and a half ago with our cat and dog and had the same issue. We stayed in short-term stay places for the first two months while we rental shopped. I basically just contacted every rental that looked like it would work for us, even if it said no pets, to ask if they would consider pets. Most said no, but a surprising number were open to cats. We finally found a place that would allow a dog as well ,provided we paid a higher bond and had the carpets shampooed when we moved out if they needed it.
The one thing about it is that you have t be open to living where ever you can find a place that will allow them. If you're set on a certain suburb it may take a while to find a place, though it might be easier with cats. When we were looking there had just been a Stuff story out about a tenant that had let their dog basically destroy the place and the courts had ruled that the tenant didn't have to pay for it, so finding a place that would allow dogs was rough.