Can a family member apply for a passport on my behalf?

No, unless you are under 18 years of age.

In particular applications cannot be made on your behalf in Ireland if you are living in Australia. Applications from Irish citizens resident in Australia must be made at the Embassy of Ireland or at the Consulate General Sydney (NSW) or Honorary Consulate Perth (WA) only. If you submit an application in Ireland a witness must verify that you live in Ireland. Submitting an application with a false declaration is a serious offence. If the Passport Office see that you live in Australia, they will reject the application.