Kill the opponent in brown body armor that appears for a short time in the following levels to unlock the corresponding weapon.

Flame thrower: Drycreek Plaza; in the store room
Gatling gun: Grassmarket Street; in the first open area
Grenade launcher; Belforte Hotel
Laser rifle: Garland Technology Center; the room with the automatic gun turrets
Missile launcher: Garland Park; among the STF trucks.