How much current can I pass through an HCT4067?
I looked through the datasheet and I can't find a maximum current.


Look again. In particular, look at section 8 ("Limiting Values"). The absolute maximum switched current is 25 mA.


  Also, I assume the maximum output voltage would be the supply voltage. Am I right?


Correct. The input should stay between GND and VCC. Higher/lower inputs will be clamped to the rails, but you must not drive more than 20 mA through those clamping diodes. (If there is any possibility of an input outside the rails, use a series resistor to limit that current.)