Why isn't the Temple Knights of Ishgard a Grand Company by now?

[Elkanah]

Rhalgr's reach vendors are a good example tbh of what they can pull off. Frankly they did that once they can do it again

The issue with this is that you DO NOT see it till AFTER the MSQ. You need to complete it. You need to kill the final primal and clear the last dungeon to even unlock this feature or access to it anyways. But it's not grand company related. It's just a vendor for your current gear. Similarly, you cannot reach diddlyshire and gear out your class before certain MSQ points or level (because it was capped at the time with tomes).

The thing is, in the Korean LL Yoshida said that they are developing something for Ishgard in the near future

Something, yes. Something developed. But it's not grand company related. They've already gone as far as interviews and live letters in the past two years, telling the community "we're not building grand companies or housing in the new area (at the time, Ishgard). please stop asking us".

Maybe one of the floors in the vault could've been the base?I'm not sure

The problem with this theory is that you would not be able to access ishgard till you're mid story or msq for heavensward. They'd essentially need to re-write quests and storylines that tie into it. And before then, the vault was closed off to WoL, the alliance and anyone in Eorzea. Squeenix already makes it clear that exclusivity doesn't mean anything to anyone, regardless of 1.0 or 2.0, 3.0 and higher. So gating it behind something that requires the story would bring even more complaints from the newer players who want to make the start there, instead of the starter areas. A lot of coding for something that they'd be better off wiping the game and starting from scratch again, to make that possible.