Plenty of campsites along the Alpine Way near Thredbo,
I have added some on the Find A Campsite part of the site.
you can keep going out towards Tom Groggin or Geehi Flats, this will involve camping with views of snowy mountains, but not camping on snowy ground. You can also do 4x4ing, motorbiking at these two locations as well.
Also, it snows most winters up near Orange, NSW, and i think you can go camping high up at Mt. Canoblas State Forest. Better to wait until you hear news of the snow falling here before planning a trip. Plenty of motorbiking you can do here as well.