The weapon store key was an item used during the Shield of Arrav quest to access the Phoenix Gang's weapon storage room. In order to obtain it, you needed to be in or have a partner in the Phoenix Gang who could 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 lost the key, you could get another one by talking to Straven. It was necessary for a Black Arm Gang member to have the key during the Shield of Arrav quest as only they could kill the Weaponsmaster.