NSW (Australian state) actually has mandatory contract tracing[1] though you can use a web or physical form.
[1]: https://www.service.nsw.gov.au/transaction/covid-safe-check
[1]: https://www.service.nsw.gov.au/transaction/covid-safe-check
Distribute free phones which is capable of running these COVID apps and then the issue comes down to just privacy.
Alternate physical form seems reasonable.
*just : Not to reduce the importance of privacy(EFF commentary in my other comment).