I got my cheap webcam and it seems to work fine.
Unfortunately I was unable to test it on the night sky because of (yes, you've guessed it!) clouds, but I tested it with my small refractor (the one I was given for the solar eclipse) on a brick wall... I could see the details of the bricks very well. They also appeared to be much more magnified than with the low-power eyepiece, so it should be nice to look at the moon and record a video when I have the chance.
(This morning I had to uninstall the USB drivers and restart the computer to make it work, but the problem is related to Windows rather than the camera, as it occasionally also occurs with other devices.)