I thought I would also chime in for those students who consider the summer as the months between December and March and because at the University of Melbourne, we recently had to figure out if we could accept students for our Vacation Scholars Program (in the context of this question a REU) from other (Australian) schools. Thankfully our administrative staff told us there was not any red tape, provided the student could live and work in Australia, and so we are happy to take applications from outside the university. Also, at least for our department, a student showing the initiative to seek out a research environment went beyond what was immediately available at his/her university was viewed quite favourably.
Finally, it is worth mentioning that the Australian Mathematical Sciences Institute also runs a summer school every year:
Again, ASMI summer school is really an opportunity for Australian undergrads.