Open your character screen & click L3 (joystick). You should notice the background gets a little darker, which means you're in "Store" mode. You can now hover over weapons & armor in your inventory & see the option to "store". You will not be able to store any ships or vehicles. You need to physically visit the vault in the tower for that. Also... when you're done storing, you'll need to click L3 again to exit "store" mode so that you can equip items.

Open the character menu and switch to Vault mode.