Here's a good map for Queenstown;
https://www.qldc.govt.nz/events-and-recreation/freedom-camping/
For Mt Hutt, it's entirely surrounded by farmland. So as long as you're in a proper self contained campervan, you should be able to park up most places just pulled somewhere off the road.
I really hope you've got chains included with your hire, know how to use them, and will be comfortable driving a campervan with them on. The Mt Hutt access road is downright hideous, and it's very long. Dirt/gravel road the entire way. Coronet peak is the only fully paved access road, and they've made some big improvements to the Remarkables in recent years, but the top section is still dirt/gravel road.