The motor still runs and I can here it when I press both the up and down directions on the switch, but the window won't move. I've taken the panel off and, from what I can see, the clips are still intact. I have no idea how to fix this.

 

I would take the door panel off then plug the switches back into the harness in the door and watch to gear on the end of the motor and see if it turns when you press the switch. If the motor is running but the gear doesn't rotate then the shaft is probably sheared OR it has a shear pin that has sheared. If the gear turns then it should move the large sector gear that raises and lowers the window. Perhaps the teeth on one of the gears are stripped.

I worked on a number of these but I've never seen a case where the motor was RUNNING but the window didn't operate. I have seen some where the gear works was jammed and the motor would hum loudly but it didn't turn. But MOST often the switch contacts are dirty and the motor doesn't get power so it doesn't do anything in one direction.