When there is an item in 1 of your 3 normal "Inventory Slots" (or equiped)
it should be possible to just drag and drop it onto the "Storage Tab" icon.
Drag-and-drop 1 of these items straight to the vault.
Right now that little vault icon can be clicked on so you can see the various tabs.
But it would be way more efficient to also use it to drag-and-drop items to Storage.
Items could then simply fall into the first available storage tab item slot.