It's a design choice more than a tech limitation. Their offered explanation satisfies the question of why we don't have a bigger Vault: visualizing such a large amount of weapons at once is beyond PS360's memory capabilities.
It doesn't, however, explain why we only have one Vault. If making a bigger Vault is not possible, then the logical solution would just be to give us more than one Vault of the current size. There's multiple terminals in the Tower and Reef and they all point to the same element - the generic Vault.
If one of those terminals let us open an Armory with all of our armor, another terminal for a Weapons Cache with all our guns, and a Stash of sorts for items we'd effectively get 3x the current space with better organization instead of the cramped mess we have now. Or just Vault 1, Vault 2 etc. with the current scheme if they want to half ass it.
Perhaps there's some other limitation they haven't told us about, but with the information they've provided so far I understand having a small vault but I still don't see why there's only one.
Then Deej's comments during E3 seem to suggest that picking and choosing is part of the experience they want us to have, and it clicked. It's not tech holding us back, it's just a design decision that follows their vision of the game. Our completionism is apparently not what they had in mind. So far it's possible, barely, to own every Exotic armor piece, every Raid armor set, PoE armor for all 4 factions, Iron Banner and Trials of Osiris sets for each of the three classes. Doing so requires you leave some stuff in the Vault though, you can't have every set on your character. If TK brings more gear than Vanilla did, then this sort of Pokédex achievement will just not be possible for multiple characters.
Weapons: several of us have every Exotic weapon, every Raid weapon, plus standout Legendaries (some Iron Banner/Trials stuff, some random drops, some Vendor items). But our inventory management is only possible with load out apps because every character and the Vault is usually just full of stuff. Passing around your "Raid load out" from one character to another involves a ton of "move to vault, take from vault, move to vault, take from vault" which, without an app automatizing it, would be beyond ridiculous. But it's still possible. With TK, our seams will burst and we'll have to make strange choices. I really can't imagine a single scenario where having to chose between keeping one weapon or another because I'm out of space would be a positive experience.
It's truly a shame. I always considered that the fact that you can eventually own every single piece of unique endgame loot in this game was a positive differentiator when compared to similar games. Seems they're throwing that out the window.