[quote]I'm on iOS for both my iPad and iPhone, and neither apps on each device are letting me transfer materials or consumables between characters or even into the vault. It says "There are no item slots available to transfer this item" despite my vault and character slots for materials and consumables having several slots free.[/quote] What you're describing does suggest that the General section of your vault is full. Are you sure that's not the case?

What are you trying to transfer? Some items have max limits you can store. Aside from that, if the item is not a piece of armor or weapon, you need to make sure there is a space available in the General area of your vault.