It is not mandatory (except when entering Tasmania) for pets to be vaccinated for specific treatments, although it is highly recommended that you consult with your veterinarian when you are conducting your health check to discuss what vaccinations may be required, especially if you are planning pet accommodation.
Treatments such as C5 for cats and F3 for Dogs are required for pets to gain entry into a pet accommodation/boarding facility and depending on your destination heartworm, flea and tick treatments may also be recommended by your local veterinarian.