If you're okay with command-line, and you're on Windows, you can try ffmpeg's "dshow" option.
First, run this command to see what DirectShow devices are available and what they are called:
ffmpeg -list_devices true -f dshow -i dummy
You will see the names of available video capture devices listed under "DirectShow video devices" and audio devices listed under "DirectShow audio devices," in quotation marks.
Make a note of the name of your device (e.g. "Cheap Video Grabber") so you can use it in the next command.
If your video capture device only appears under "video devices", try this command:
ffmpeg -f dshow -i video="Cheap Video Grabber" outputFile.mp4
If your video capture device appears under both "video devices" and "audio devices", you may need to specify that you want to capture both streams:
ffmpeg -f dshow -i video="Cheap Video Grabber":audio="Cheap Audio Grabber" outputFile.mp4
If you prefer AVI instead of MP4, just change "outputFile.mp4" to "outputFile.avi". There are dozens of other supported formats.
If you're still having issues with audio/video sync, ffmpeg has an option to let you introduce a delay into the audio or video stream to get them back in sync.