How do you enable tap-to-click via command line?
Xmodmap is of no help here. Xmodmap controls physical-to-logical mappings of buttons and keys, not the physical process that generate events in the first place.
The tool for this kind of options is xinput. The property name depends on your touchpad model, it may be something like
xinput set-prop "SynPS/2 Synaptics TouchPad" "Synaptics Tap Action" 0
Run xinput list to see the names of available devices and xinput list-props "the device name" to list properties of a device.
See also Dynamic Input Configuration with xinput on the wiki, and some examples.
You can use gsettings:
gsettings set org.gnome.desktop.peripherals.touchpad tap-to-click true
Which enables tap to click.
gsettings set org.gnome.desktop.peripherals.touchpad tap-to-click false
Will disable it. This is the same as changing it in System Settings.
Based on the main answer, given that I had to modify some of the instructions there:
The command to enable tap-to-click is therefore of the form:
xinput set-prop "device" "action" 1
To read the "device" you have to do
xinput list
But it may prove difficult to identify the device in that list. Some tips: it is probably under "Virtual core pointer"; it may contain terms like "Syn", "Synaptics", "Touchpad", "Alps", "Glidepoint". e.g., mine was AlpsPS/2 ALPS GlidePoint id=16, but I had to guess; as I was not sure I have tested if that was the correct ID number by disabling/enabling the touchpad with xinput --disable 16 and xinput --enable 16.
Now, to get rid of all the confusing names, ID-numbers can be used instead of the device and action names.
So, to read the "action":
xinput list-props 16
Which listed among others:
libinput Tapping Enabled (297): 0
So, using ID numbers instead of names, the final command was:
xinput set-prop 16 297 1
Note: for some reason, using the name of the action within the command, as suggested by the main answer, wouldn't work for me (xinput set-prop ""AlpsPS/2 ALPS GlidePoint" "Tapping Enabled" 1), while using just the name of the device did work (xinput set-prop "AlpsPS/2 ALPS GlidePoint" 297 1).
This command can be useful in systems where there is no GUI for such setting, like in the LXQT that I was testing at the date of the post.