On this page of our game guide to Batman: Arkham Knight you can find a description which explains how to destroy Penguin's weapons cache. Batman has to use an explosive gel to destroy all weapon supplies.
Enter the vault, approach the weapons cache and select Explosive Gel from the inventory. Spread the gel there, but don't detonate it yet, as the explosion would kill Batman. Instead, walk out of the vault and close the large door. After doing this, press the detonation key/button. You will unlock a new side quest connected with other weapons caches. Use the grapnel on the balconies on the top and listen to a conversation with Alfred.