Welcome!
You can always try the major airlines, Qantas and Virgin Australia, although getting into their apprenticeship programs is very competitive. Other than that, there are regional airlines and GA operators that may take you on. I'm in Victoria, so the majority I know of are from Vic, but you may like to give them a call. I'll list where the apprenticeship would be based too.
Kestrel Aviaiton (helicopters) - Mangalore Airport
Qantas - Brisbane
Virgin Australia - Melbourne and Brisbane
Qantaslink - Tamworth
Field Air - Ballarat
Avionics 2000 - Essendon
Execujet - Essendon
JetStar - Newcastle
AG Airwork/AG Flight/AG Aero - Stawell and Deniliquin
Rex - Wagga Wagga
Hawker Pacific - Cairns
Asia Pacific Aircraft Storage - Alice Springs
Note, I can't guarantee that all, or even any, of these companies will be looking for apprentices. However, it's worth giving them a call.
Another way to hunt for jobs would be to search for regional airports around Australia, look up their business directories then cold call each company listed. And best to call, rather than email. Emails get lost too easily. Explain your situation, what you're after and if you can send them a resume. Then cross fingers, toes and everything else and hope.
Best of luck and keep us updated!
Cheers