I believe many will work, but I don't believe the cost is particularly low -- I know Amaysim will allow roaming internationally, but at a cost. Vodafone has both an Australian and NZ presence (although I believe they operate independently), so perhaps you'd have more luck with them?

I would think the cheapest option if you're planning on using service with any regularity in both countries would be to buy a SIM card in both places. But that's just a guess -- I could be way off in that regard.