But, there is hardware involved too. If the webcam has not been designed to be able to operate with the LED off, then it can not be exploited.
Firmware controls the hardware, and those aren't hack-proof either. Then again, there may be webcams where the same switch activates it and the LED. But we know that's not always the case. So why take the risk when black tape is 100% reliable?
I wouldn't trust using a cover or a piece of tape, not with cats who like to swat small objects and pull on loose items like string or tape. Without the source code for the cameras software and a wiring diagram for the device itself, how would you know either way?