UPDATE 7/24/17: This was fixed with the Season 13.5 patch on 7/18!
I played The Vault with a toon that has reached the Ship Slot limit (I'm counting 61, and the C-Store won't let me buy more). I have plenty of inventory space in all categories: personal, bank, account bank, active ship slots, dry dock slots, etc. The only thing that sets this toon apart from my other toons is that Ship Slot limit.
When I reach the part of the mission where I talk to De Soto and am supposed to get a requisition for a Type 8 Shuttle (it's a Fed toon), I see an "Inventory Full" message. I see no Shuttle Requisitions at the Ship Requisition Officer, and there is no Type 8 Shuttle available. Also, every time I have rezoned since this occurrence, I see the "Inventory Full" message. I've tried logging out many times and dropping The Vault and re-doing it, but it's still happening.