To reproduce:
1) Open attached project and scene
2) Build and run for WebGL
3) Play and stop Webcam on Browser A
4) Keep tab on
5) Open Browser B (no repro if keeping in the same browser and just getting new tab)
6) Try playing - browser asks camera rights, but doesn't use them

Note: If the Browser A tab is closed, don't even need to restart Browser B tab's app to gain Webcam's control, clicking Stop and Start is enough to make it work

Tested with Firefox and Chrome (both as browsers A and B)

Reproduced in 5.4.0f3, 2017.1.0b1