After stripping any original track of its vocals would it be possible to have the original vocals only be playing back in a wannabe Elvis’ ear on an exclusive track that is not broadcast.
A revolution in karaoke, any original music may be used for a karaoke then, all it needs is software to rip the vocals on the fly and prompt/play them only in the ear via a special third track, words as usual still on screen as well.
IF you have both the karaoke track (no vocal) and the solo vocal track, then you could make a stereo track with the karaoke version on one (example, the left) channel and the vocal version on the other channel (example, the right). Using the example channel allocations, you could connect the left audio output from the computer to the PA system and the right channel to a headphone amplifier.
However, the Vocal Removal effect in Audacity will only remove (or reduce) the vocal in certain cases. Often it won’t work at all, and the effect cannot isolate the vocal. (see here: Missing features - Audacity Support)
So in answer to your question, yes it may be possible, but not without either first obtaining the karaoke and vocal tracks, or using tools other than Audacity.
Given the above reply, that is not relevant to Audacity software.