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.
We're an apparel company and were looking to set up a print-on-demand facility within the next 3-4 months for our own internal designs, but perhaps make this available to other companies also.
If you could email me and provide more information such as your estimated quantities and requirements, perhaps I could help you.