Shifter won't click and cable doesn't move

It's likely a very simple issue - the cable broke. Possibly the head of the cable detached so that the shifter cannot grab the cable. There is a remote possibility the shifter itself broke where it holds the cable.

You can verify this by detaching the cable from the derailleur, you may find that end of the cable is loose and can be pulled out of the housing. Find a manual for your particular shifters and find out how to remove the cable. You may need to remove a cover to get access.

Yo should also verify the derailleur can move by pulling the cage carefully outwards - it takes a bit of force. The derailleur is robust enough that you will not break it.

The likely reason the shifter is not clicking is that it relies on the derailleur spring pulling on the cable for tension, so the internal parts are not moving when you work the thumb and finger levers.