Can one apply for a work visa in Thailand upon entry with an APEC Travel Card?

Unless I am mistaken you cannot use an APEC entry stamp to apply for an in country Non-Immigrant type visa.

The visa you want is a Non-B visa based on business and issued for employment.

You can get documentation from the employer and apply for it in your country before you come (which you could do via the online eVisa website thru Canberra or Sydney depending on where you live in Oz)

OR

you could come in without using the APEC card entry (free 30 day entry stamp) and then apply for the in country 90 day Non-B visa at the immigration office where you live/work

hopefully other APEC card members will weigh in on if you can or can't apply for the 90 day Non-B in country from that entry stamp

Or you could go to the immigration office where you'd apply and ask or even ask your future employer they certainly would know.