As I understand it, my wife will need a EEA family permit to come to the UK with me.
This is incorrect. As an Australian national, she does not need a visa to enter the UK, so she can just fly with you to the UK and present there the same proofs that would normally be required for an EEA family permit application. She can even fly to the border by herself and present those proofs, if you are in the UK when she does so.
Is there a way for us to get the EEA family permit prior to our departure for the UK?
Yes. In fact, you can only apply for the EEA Family Permit when you are outside the UK, because it is the functional equivalent of a visa.
Once you are in the UK, whether you've arrived there with a family permit or not, your wife will want to apply for a residence card of a family member of an EEA national.
I seem to need something called a EEA National Registration Certificate number, which I need to be physically in the UK to obtain.
For the EEA family permit application, and even for the residence card application, you do not need an EEA National Registration Certificate number; the EEA National Registration Certificate is an optional document. See the relevant page on gov.uk for more information.
If the entry is required on the web form, respond with "N/A" or "none held" or some such.
Although your wife does not require an EEA family permit, she may wish to apply for one anyway. There is no charge, and, among other benefits, the family permit application may be less stressful than just applying for entry at the border. When you apply at the border, you are likely to be tired from traveling and under stress because of the possibility of being refused entry.
In addition, if your wife wants to look for work early on, having the family permit could make that easier. Without it, she will have to wait until the residence card application has been filed.
Finally, a tip: when you travel with your wife to any Schengen country, you can use the "EEA/EU/CH" passport lane even though she does not hold such a passport, because when she travels with you she is a "person enjoying freedom of movement." This is true even without the EEA family permit or the residence card, though strictly speaking without those documents you could be asked to prove that you are married. In practice, though, I doubt that you'd commonly be asked.