An Estonian ID Card reader otherwise known as a Smart Card Reader can be purchased in Australia through the following channels: They usually range from $15-20.
It appears that Officeworks do not carry them.

I live in Australia and don't permanently reside in Estonia, why should I use an ID card?
  • Vote in Estonian elections (I-voting)
  • Manage/Withdraw/Transfer funds from Estonian bank account in Euros
  • Fully renew your passport and/or ID card electronically
  • Send and receive documents securely (digital signatures)
While visiting Estonia, say as a tourist, you may use it...
  • As a legal travel ID for travelling within the EU
  • As identification on Estonian public transport
  • As a driver's licence identification / renting a car
  • Taking out an Estonian SIM card
  • As a national health insurance card
  • Visiting local doctors/hospitals and using e-prescription services