What 3PP and merchant account providers do people know that allow merchants to accept credit cards over the phone?
My business operate in Australia and mainly deal with Australian customers however, my company is incorporated in the US (Delaware to avoid massive tax burden).

I'd like to avoid large monthly fees as our volume is probably going to be extremely load and definitely need a reliable company to work with.

Welcome to the forums malagasy!

You might check out Payquake - they might have an option for you down under. If you did have a US bank account and US address, you could qualify for a US merchant account actually. Or you might check out http://www.firstdata.com.au/

For starters you can acquire Domestic MOTO merchant account in the US, since you have an incorporation. Or you can also go with a MOTO EU merchant account. Your discount rate and transaction rate will be a little higher, so I would advise going with the US account first, and if all else then go for the EU.