How/When can I access bank vault?

[Enkidoh]

Not specifically, a retainer is like a storage NPC you can hire, the ability to do so is unlocked automatically as part of the storyline (specifically, around level 17 when you've

completed the first three story dungeons and joined the Scions of the Seventh Dawn).

Then speak to a retainer vendor in the Market area of each city (identified by a two faces icon on the map). You'll be asked to create a race, appearance and name for the retainer, which is just for aesthetic reasons. Then to use one you just find a 'summoning bell' to call them.

You can hire up to two retainers per character so they're well worth using.

Otherwise the closest to a 'bank' you're thinking of is the Company Chest that a Free Company has - the ability to use it is unlocked in a Free Company once that FC has reached a certain level. To access it, click on the gold coloured cupboard labeled 'Company Chest' that is found in both each city's Grand Company HQ, and also in the Market area of each city (usually the later location has a retainer Summoning Bell nearby too).

The founder/leader of the FC also can dictate who has access to the Company Chest as it's not a personal bank, but a communal one, shared by all members of that FC. But it's worth still using anyway if you are a member of a FC - it's one of a FC's many benefits.