So the basic answer is "sure why not?" with the caveat that your recording will likely include a low-fidelity copy of the instrumental by way of the computer's speakers and microphone.
Just did a quick trial and this method definitely works. Possibly could separate the instrumental from the vocal through use of headsets. Will explore further tomorrow ... thanks for suggesting this method!
A variant of this recording method might be to display a metronome on the stage to control timing / pacing while recording; then play back the vocal track in parallel with the instrumental.