Just being right is wrong.

In both of my DAWs: Audacity and Reaper, when I try to record from any service on the web, it only picks up the right track. That’s when I have Stereo set in Audacity. When I switch it to Mono, I don’t get anything. Since both DAWs are affected, I suspect my computer’s soundcard (Acer quad-core with Win 10). Any suggestions to check this?
If I play a Left speaker/Right speaker test, I get the Left in the centre, as if in mono,… and the right does not sound at all.

Are you using [u]WASAPI loopback[/u] as your audio host & recording device?

Nope!

WASAPI loopback should accurately “capture” the digital audio stream and your soundcard is not directly involved.

If I play a Left speaker/Right speaker test, I get the Left in the centre, as if in mono,… and the right does not sound at all.

Is that a playback problem? i.e. Does it happen with known-good audio files or when you’re listening to streaming audio? That could be a hardware problem with your soundcard. Or it could be your speakers/headphones. Do you have something else to plug-in? You could also get cheap USB soundcard for testing purposes. I keep a couple of inexpensive (but not super-cheap) USB soundcards around just for testing/troubleshooting.

Maybe recording is OK? Do you see the left & right waveforms in Audacity, or only the right (bottom) waveform?