The only medical and dental schools that require you to submit a separate ‘written application’ in addition to applying through the normal admissions center are James Cook University (JCU) and the University of New South Wales (UNSW). This is explained in more detail in this article.
For both JCU and UNSW, the written application for 2022 entry is due on 30 September 2021, although you’ll need to start much earlier if you want to write up a good application and also submit it on time. (For a full list of the key dates in the medical/dental entry process, refer to this page.)