Silver Bullets originated after the creation of firearms during the 17th century; which was the time when they utilized silver tipped arrows against Werewolves, Vampires and Demons. When using firearms on the modern battlefield; people began to realize that bullets could be created out of Silver.
When being introduced in the war against the dead and damned; these bullets were proven to be the most effective weapons; alongside that of Silver Stakes and other weapons.
To this very day; silver bullets have been utilized by many factions over the world. The Vatican has utilized them; as well as the Black Beret Corps. Alpha Wolf is known to have a ton of these at his house; just in case he's ever attacked by Vampires, Demons, Werewolves or anything else that can't stand silver.