Apparently, xf86-input-synaptics needs a restart to get the touchpad tap functionality. I restarted X, but that wasn't good enough.

PS—No, it just quit working, so I have to figure out what this means.

 

“[2016-06-18 18:53] [ALPM-SCRIPTLET] >>> xf86-input-synaptics driver is on maintenance mode and

[2016-06-18 18:53] [ALPM-SCRIPTLET] xf86-input-libinput driver must be prefered over. ”

I installed it, so maybe it will enable tapping after restarting X.