Controlling an LED with the 74HC4067E multiplexer
Pin 15 on that MUX is the enable pin, you need to tie it to ground. Otherwise, it's just floating, and it only would have worked by accident.

Also, you state:


  I am also providing 5V to the Vcc pin on the MUX. I don't really know why I am doing this, but it seems to work.


The chip needs power to work. The Vcc pin is where you give it the +5V power. Without that, it might still work, but only by accident and depending on the inputs on the other pins.