The Vault should be an options menu page available in orbit or in the Tower.
The UI would be close in appearance to the Character page but with an item grid between the gear category icons instead of the character model.
When you open a gear category: Primaries, Specials, Legs, whatever - it stays open.
To open another category you hover over its icon for ~ 1/4 second and it opens and the previous one closes.
When a category is opened the grid in the middle is populated with all the items of that category that this character is able to equip.
Click on an item in the middle to select it.
It is put into an available equipped item slot but also stays in the middle (it is drawn dim to indicate that you have already got it).
If all available equipped weapon slots are full then all middle items are drawn dim.
Click on an equipped item to un-equip it.
Notes:
- equipped items remain in the same place in the middle grid so their location persists for a long time and can be memorized so you can find items easily
- because they remain in the middle - multiple characters can equip a single weapon without playing the Vault-Shell-Game mini-game.
- you equip your one-and-only Gjallarhorn once for each character and then never have to touch it again - no more which-character-has-it? questions - no more shell games - no more making your fireteam wait whilst you get equipped - equip it once and leave it alone
Also:
You should be able to switch characters in orbit or in Tower as easily as you change shaders.
The character banner on the top left of all options menu pages should open as a dropdown menu that shows all your character banners. Select a banner to switch to that character.
It should be the player who is a member of the fireteam - not the character. Therefore switching character shouldn't force you to leave a fireteam.
How do items get in to your Vault?
The gear grid in the middle shows all gear owned by you, the player. So whenever any character picks up a weapon - it is appended to the Vault.
If your Hunter deletes an item that is equipped by your Titan then it become un-equipped for the Hunter but the Titan still has it and it is still available in the Vault.
An item only truly disappears when no character has it. And there will be a waste basket to catch accidents.
So if your warlock picks up a rare shotty and you don't want it then you just delete it at the first opportunity. No other character will have equipped it so it will just be deleted. The new Vault's ability to share gear between characters will not interfere with discarding dross.