The Audio is not recorded when I capture an AVI file using the WinTV capture program.

It is actually the Sound Card that will capture the Audio portion of the AVI files. The Win/TV capture application will allow you to capture both by syncing the Audio and Video together. But the sound card will actually capture the sound.
If you do not record the Audio when you are capturing Video Clips, you want to make sure that the sound card's Line In mixer is setup to Record audio. Sometimes they are set for Playback only. To Check this, click on Start - Accessories - Entertainment (or Multimedia), in the Windows Volume Control, go to Options then Properties. In here click on the Recording Bullet. Below you should see the Mixers that are set to record make sure that the Line In is checked and click on OK. Then make sure that it is not muted. Try to record some audio through the Line In jack using Windows' Sound Recorder. This program is found in the Multimedia group.
If you can record audio using the Sound Recorder, then you should be able to capture audio with the capture program.