Weapons are automatically added to the Weapons Cache when they are picked up or purchased, but new weapons are not displayed in the Weapons Cache until after switching to another weapon. Picking weapons up on the street automatically adds them to the Weapons Cache, and there is no need to go to the Weapons Cache to "deposit" them.
The full Weapons Cache is accessible through the Store menu at either Brass Knuckles and Friendly Fire. Instructions to access the Weapons Cache are displayed at the bottom of the screen. Due of this, the Weapons Cache is available before unlocking the first Crib in each game.
Many weapons used in missions and activities are not retained after the mission ends. For instance, the weapons from the Prison Armory during "Jailbreak", and the weapons used during the Fuzz and Drug Trafficking Activities. In general, if the weapons are unlimited, or are weapons unavailable outside of the mission, they are not added to the Weapons Cache.