You will need an Australian Open license for a car, or a Provisional License for a car. However, there are mobile phone restrictions you need to know about. https://www.qld.gov.au/transport/licensing/driver-licensing/applying/provisional/restrictions#mobile
This means that if you fall into the P1 category, you may not be able to perform in your role as a Pet Taxi Driver if a Clinic or a Pet Owner needs to phone you for pick up or collection reasons.
For more reading on license types, head here: https://www.qld.gov.au/transport/licensing/driver-licensing/types