Is there any software, preferably free, that will enable me to turn my iPhone into an audio recorder?

no you would need a piece of hardware that records it. There’s loads of stuff out there, so look in an apple shop or something.

sure, but what about the mic in the earbuds?

Free app called vNotes. You’ll have to jailbreak the phone and download it through installer app.

I have it installed myself, great little app.

I second Spargett. Jailbreak the iPhone and you get access to the free program called vNotes. You can record voice, and also copy your voicemails with it and transfer to your computer.