It's not about being allowed to have bankai. It's a fact that none of the current vice captains has it. And bankai is supposed to be extremely hard to achieve that it takes someone to rise far above the level of vice captains to achieve it, hence why Renji is an (terrible) exception.
Ikakku doesn't count since he's been captain-level for quite some time now and was only refused of the VC position because Zaraki and Yachiru are simply inseparable. And before anyone says it, Ichigo cheated so fuck him.