Does a woofer have to sit on the floor or can it be on a shelf?

You would probably be better off with it sitting on the floor, as it could possibly vibrate during heavy periods of bass sound if placed on a shelf.

If you wanted to place it on a shelf, you could always reduce the bass output, but in my opinion, this would negate the reason for having a sub-woofer in the first place, and could lead to the system sounding rather tinny.