Yes, you need a minimum of a Motorsport Australia Circuit license to race a Formula Vee in circuit races in all states. To obtain the license, you need to pass a health test, learn basic safety rules such as flags, attend a one day driving observation day in your race car or similar. Check the Motorsport Australia website for more details. At this time, a CAMS PCC license is $455 per year. The medical exam will typically cost about $150 depending on which doctor and practice you attend, and the driving observation day will cost around $500 depending on what you attend. The entire process should take around 6 weeks.
Some tracks in some states use a different license system, but generally your Motorsport Australia license will be recognised. Check with your state club if you are unsure.