The Swing magazine is a virtual magazine that is required to throw melee weapons like the Axe. It does not spawn in buildings and cannot be bought at traders. Once you equip an axe, it just magically appears.
Weapons are only functional in Arma if they have some ammunition. If there is none, the player cannot "fire". Each melee weapon is basically a normal gun with a limited range. So they are kind of a work-around to implement melee weapons.