You need to use each of these weapons to get at least one kill on an enemy (civilians may not count, best to get the kills on enemies - though apparently dock workers will count). I've listed the shops you can buy them at, but keep in mind some of these weapons can be picked up off dead enemies.
Machine Guns (Up on D-Pad):
- Thompson M1A1 (Not sold in shops - see note below)*
- Thompson 1928 (H)
- MP40 (H)
- Grease Gun (H)
- .38 SMG (H)
Handguns (Right on D-Pad):
- Magnum (H) (G) (N)
- 1911 Special (G)
- 1911 Semi (N)
- C96 Pistol (H)
- .38 Revolver (N)
Shotgun/Rifles (Down on D-Pad):
- Shotgun (H) (N)
- M1 Garand (H) (N)
- Kar98 Rifle (H)
Throwables (Left on D-Pad):
- Grenade (H) - (See note below)**
- Molotovs (G)
(N) means the gun is purchasable from a normal gun shop (all over the map, icon is a handgun). You can also kill the shop owner and steal the guns if you like.
(H) means the gun is purchasable from Harry's (top left of the map in Kingston, icon is a machine gun).
(G) means the gun is purchasable from Giuseppe's (near the middle of the map in Little Italy, icon is some lockpicks).
* The Thompson M1A1 (the one WITHOUT a drum barrel) is carried and dropped by some enemies during missions, and cops also carry them when you have a 4 star wanted level (thanks Calex dEUS).
"In the "GAI Truck" car jacking mission, one of the soldiers guarding the truck has a Thompson M1A1." (Credit to Protium)
** Make sure the grenade gets the kill, not a secondary vehicle explosion.