Yes indeed. The difficulty mostly lies in departments that require what’s called “baseline security clearance”. Even us NZ citizens aren’t eligible for that - need to be Australian (the reverse does not apply in NZ). I’m with a state government department right now which is a lot less complex.
I’m not saying that a petition won’t work (though in my experience with petitions I don’t think it will), but if there is a senator willing to go to bat (that isn’t Pauline Hanson) then I think you’ll have much better luck working with them than a petition that the government (represented by Tudge and Dutton who we know are horrible people) can just ignore.