Create a another track and set it's input to Virtual MIDI Keyboard, record arm & monitor the track.
Right click it's record arm button and set it to Record: disable (monitoring only)
Make this track a child of the mic track
Right click the record arm button on your mic track and select Track Input FX Chain
Place ReaEQ on the track and drag the volume slider on the right all the way down.
Toggle the bypass (top right corner next to wet control) on and off
Go to Param, which should now say Last Touched: [Bypass], and select MIDI Link > Note > 60 (c4)
Set Baseline all the way to the right and scale all the way to the left
https://i.imgur.com/ILmuMhO.png
Enable the onscreen virtual keyboard ("View: show virtual MIDI keyboard" in actions list)
Right click in the blank gray space of the MIDI keyboard and select "Send all keyboard input..."
Locate C4 and right click it
Now, as long as that keyboard is open, the Q key is going to mute your microphone input.
Yes, it is a bit sadistic.
Short method
Own a MIDI controller of any sort
Use ReaLearn extension for Reaper to set any MIDI key, CC message, etc to momentary switch for the FX bypass