Sure! A couple of ways. I a haven't tested them but.
1. you could plug the line out from your sound card to a cassette deck, and record the whole sortie.
2. play back the cassette with the cassette deck plugged into the stereo line in of the sound card, while recording that source with Cool Edit Pro, Sound Recorder, Sonar, Sound Forge, or any other decent recording software.
3. Save the file as an mp3 file (Cool Edit Pro can do this--that's what I use to create mp3s)
Now, you might be able to record the sortie using Cool Edit Pro at the same time you're flying, but that's what I haven't tested. Also, I'm not sure how running recording software at the same time you're flying would affect performance in AH. That's why the cassette (or other external recorder) is the most foolproof way to record the audio.
Oops. I see you want only voice. About the only you can do that would be to leave "effect" volume up and turn down (all the way) the other sounds via the AH setup menu. Kind of hard to fly that way though. Maybe you could fly along with the sortie at high alt with your job being to record the voices, without having to worry about mixing it up with the enemy.
Cool Edit Pro allows you to take a snapshot of a section of audio and then remove all of that type of audio from the rest of the waveform. So, your snapshot could be of your plane's engine sound. So you might be able to fly with all sounds, and then remove most if not all of the engine sound later (if you have CEP). Removing the engine sound might also affect the quality of the remaining audio, since many of the frequencies that made up the engine sound might have been frequencies that were part of the voices, etc. That's a trade-off you'd have to live with.
Once you get all this worked out you could put the mp3s up on a free OMD site (online music delivery, mp3.com, etc.) in the spoken word genre, so that we can all listen to it.
