It's possible to do just about everything in single player (aside from Electron Mace and Riot Lance it seems), but if you have the option to play online, coop is a way, way better choice to efficiently get kills for these weapons. You will probably take care of most of the kills for these weapons during single player, the rest of which you can mop up during coop.
- TAR-39 Assault Rifle
Found throughout the campaign. Loadout weapon in coop.
- G290 Minigun
Found a few points during the campaign with unlimited ammo. Reactive armors carry them in coop.
- CQC11 Shotgun
Cayman Global troops carry this in campaign. Loadout weapon in coop.
- RXL-240 Electron Mace
Aspari sergeants tend to use this in coop mode. Colorado will have a few, so will China and Scandinavia.
- COIL Laser Rifle
In the last few levels of campaign. In coop, sergeants and above will have them in Colorado, China, Scandinavia.
- LTB Thermite Flamethrower
Found in Downzone during campaign. Reactive armors will have them in coop. Can be found in coop in wall weapon boxes (like the first ammo box room of Colorado on hard).
- Razorback Revolver
Can be found in Downzone in campaign. Loadout weapon for coop.
- Fragmentation Gernade
Loadout weapon. You switch to it by holding . You get them in the campaign mostly by disarming enemies, but in coop you always start with a full load of them.
- APP3 Pistol
Found throughout campaign. Loadout weapon for coop.
- SWARM Rocket Launcher
Found during the first chapter for Cayman Global. However, grinding kills with it is inefficient there. In coop, the SWARM can be found on Argentina and New England in wall boxes, and dropped by sergeants. 50 kills is more than manageable with the SWARM in coop in just one run, if you find one.
- UMK MkIV SMG
Found in Downzone for campaign. Loadout weapon for coop.
- Kusanagi HSR-6 Sniper Rifle
Can be found in campaign sparingly. Loadout weapon for coop.
- LAWS-92 Rocket Launcher
Rare in campaign. In coop, on hard or above, they are all over the place. Western Europe on hard alone has 4-6 easily obtainable ones being held by sergeants.
- Kusanagi ACR-10 Assault Rifle
In campaign, Cayman Global carry these. Loadout weapon in coop.
- EMW-56 Gauss Gun
Can be found a few times in campaign. In coop, Eurocorp and Aspari sergeants will have these.
- Bullhammer Mark II Revolver
Downzone in campaign. Loadout weapon in coop.
China and Scandinavia are the best levels to get this on coop. The cops will drop these throughout the level. If you haven't upgraded the weapon, you might want to wait until you do; it's rather weak, and it's alternate fire really helps in getting kills.
- Mjolnir H.O.G. Assault Shotgun
Cayman Global sergeants drop this in coop. Western Europe on hard is a great level to farm this one.