switches - Replacing a mechnical switch with a digital one

I have a circuit which allows me to manually (physically) switch between two VGA inputs (PCB push switches) to feed one VGA output (a standard commercial KVM). I want to be able to switch between the two inputs digitally - ie. from an output of my arduino.

I'm happy with everything I need to do from the arduino side of things, but I don't know how I should go about replacing the mechanical switch with a transistor (or even if that's the right thing to use!)

How should I go about making this swap? Thanks!

EDIT: Some photos of the KVM for your viewing pleasure!