The Weapon store key is an item used during the Shield of Arrav quest to access the Phoenix Gang's weapon storage room. In order to obtain it, you must either be in or have a partner in the Phoenix Gang who can get the key.
This item was previously named just "Key" until the Varrock update, when it was renamed to "Weapon store key", possibly to avoid confusion with other keys. However, this led to scams where the key was being sold for tens or hundreds of thousands of coins.
If you lose the key, you can get another one by talking to Straven. It is neccessary for a Black Arm Gang member to have the key during the Shield of Arrav quest as only they can kill the Weaponsmaster.