The Australian Vaccine Claims scheme covers losses or expenses of $1,000 and above due to administration of a TGA-approved COVID-19 vaccine, or due to an adverse event that is recognised to be caused by a COVID-19 vaccination.

The TGA has an adverse reporting system for the 5 approved C-19 injections here:

Database of Adverse Event Notifications (tga.gov.au)

https://apps.tga.gov.au/PROD/DAEN/daen-entry.aspx