You can apply to bypass the requirement of his consent through the court, but you first have to attain a s60i certificate which says you have attended a family dispute resolution conference to try and reach an agreement about the issue.
It's interesting that this is a dispute between parties who are still together. It's very uncommon. Realistically, I do see some challenges in attaining an order allowing the kids to travel abroad without the father's consent.
If the relationship is troubled, he may be concerned that you will leave the country with the kids and not return. Where a parent has family in the country they are wishing to visit, it may also give credit to this concern because it means you have a support network if you were to relocate internationally. If you don't have a job, as well, the court may see little motivation for you to return to Australia which again heightens the perceived risk of relocation.
To overcome these risks, the court might order that a bond be paid into a trust fund to legally aid the father in pursuing recovery of the kids if they are not returned to Australia.
They might also refuse the request and place the kids on the airport watch list.
It's impossible to predict an outcome, of course, but first step is organising family dispute resolution. Legal Aid or Relationships Australia can help.