Whilst looking through a couple of posts on the slickdeals forum I came across this thread that listed sites that give away free stickers.
Anyway, I ended up making the following list of sites that accept Australia mailing addresses (and I added a brief description of the company or site to help you choose what stickers you want).
Some of these are quite old, so not sure if you'll actually receive anything but hey, there's no harm in trying - you'll just need to enter your details (& use that spam email address that you have):
89.1 WBOI Northeast Indiana Public Radio
Burton Lifestyle brand (skate, ski etc)
Care About Climate Climate change volunteer group
DigitalOcean Cloud computing company
Element Skateboarding-centric apparel
Elite MMA Shirt Club MMA stuff
Fair Chase Hunter Something about hunting
GoFlight Flight Simulator Controls
GoPro Has been posted before by EC and a few other posters
Gunnars Specialised eyewear - particularly for gaming
HNGR Foodtruck APP
JustStand.org Standing furniture website 'front'
Mosko Moto Travel bags for motorbikes
PaintballRuinedMyLife Paintball stuff
Punisher Skateboards Skateboards
Smileround Smiles
Soil Science Society of America People who like clean dirt
Starboard Sups Stand Up Paddle boarding company
STICKER ALBUM!