To unlock a weapon all you have to do is pick it up, it automatically becomes available at any supply point. Just need to keep an eye out for any weapons you don't recognise.
Here are the 10 weapons I had at the time of unlock.
These ones are the "normal" weapons I found at the Red Security objectives
Pistol - First Weapon you get as soon as you spawn
PDW - Very common, found on most Security enemies
Shotgun - Found a few used by troops in precincts.
Conqueror - Again, found in precincts and hardpoints. Used by the Nova Troops
Machine Pistol - Like the others, quite common and used by basic nova troops.
Omni Rifle - Found one being used by a larger enemy at one of the hardpoints early on.
Homing Launcher - Found one on a blue pedestal in Khan's Scar at the Forge Hammers objective.
Cyro Shotgun - Found on larger troops
Vortex Cannon - Unlocked at Firearm level 3
Minigun - Stationmaster "Charon" dropped it
These are "Sci-Fi" weapons at purple Logistics objectives
Plasma Rifle - Common at Monorail stations, used by robots.
Arc Rifle - Got this from the first boss of the game I believe
Pulse Beam - From one of the Monorail robots you have to defeat to complete the objective.
Industry weapons found at a Green Industry Objective
Mulcher - Found by a larger enemy at Chimera Field: Founders Footsteps
Bolt Thrower - Found at Machinery in Khans Scar
Mass Driver - Khans Scar again, at the Forge Hammers to the west
I will update the list with other weapons I find