Hi Clayton and welcome to the forum
If a supplier requires a tax id to open an account, then a local NZ tax id should do just fine. All you have to do is tell them you're from NZ and that will be fine. Suppliers require a tax id for their own tax obligations.
Now all wholesalers will require and the larger dropshippers will also require, but some smaller dropship suppliers may not, so if you don't have a tax id and don't want to get one, then I would suggest digging through and trying to find a dropshipper that doesn't have the tax id requirement.