We don't need more vault space (but it couldn't hurt.) We need shared inventory across all our characters.
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Isn't that exactly what the vault does?
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When you go to your character screen and choose a weapon, you can only choose weapons that character is carrying. To switch guns between characters you have to put the items in the vault, then pick them up with your next character. A shared inventory would mean the vault would only be used for trophies. Every character would have access to the same inventory screen. You wouldn't have to put something in the vault first, then pick it up with the new character. It would already be in your inventory.