The Microwave Cannon (or X-43 MIKE as it's called in-game) is a close-range, high-damage weapon that is particularly effective against Ceph.
The earliest opportunity that you have to find on is in Mission 9, "Dark Heart", in a subway tunnel shortly before reaching the mission's main objective. The MIKE comes with 500 ammo and kills quickly, however you can not replenish its ammo at ammo boxes.
The small, insect-like creatures scattering about through the subway tunnels in this mission count, so kill as many of those as you can with the MIKE and otherwise keep it on hand for Ceph Stalkers. You can get all 20 kills needed within a single MIKE.