Kerstin here from the Shopify Guru Team :)
We have a lot of T-Shirt fulfillment apps in our App Store and you can find a list of many of them here in this earlier discussion. However, there doesn't seem to be an Australian based app just yet but you can use Ogo to integrate T-Shirt dropshipping with your Shopify store. Ogo now have a 'private app' that you can use to link the service to Shopify once you sign up on their website. Similar to the many other T-Shirt dropshipping apps, Ogo also handles the entire fulfilment process and ships direct to your customers Australia-wide.
I hope this helps. You can respond directly on this thread, or via email for any clarification.