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- Dominion: Seaside
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- Dominion: Prosperity
Vault does best with big handsizes, and Caravan is just about the cheapest non-terminal route to big hands. Of course, Laboratory, Village/Council Room, and Tactician (among others) are also great ways to increase your hand size, but they’re much more expensive and therefore less viable in poor decks. In order for this to work, you have to really spam the Caravans, which is not too difficult since it’s relatively easy for a deck to consistently hit $4.
Haven can substitute for the Caravan, but it’s less readily spammable than Caravan because each Haven you play lowers your hand size. It can, however, move your Vault from a small hand to a bigger one.
I won this game for two reasons: first, my Caravans were more effective than his Caravans, because drawing even a Curse meant another +$1 for the Vault; two, his Worker’s Village/Rabble combo was slow in the beginning with Curses and slow at the end with Victory cards, while Vault can keep on churning even when the deck is clogged with Curses and Victory cards.
Dominion Strategy 
Haven also increases hand size, on the turn after you play it.
You’re very right. Included a mention.
Native Village combos well with Vault, for the same reason. A string of Native Villages (they’re cheap, you can get them in large numbers) can pick up cards for you that you put into your hand when the Vault appears (either on the Native Village mat or in your hand). This works best if you have a +buy card available as well.
I have found Vault to be extremely effective along with Tactician. Which means you can get off a solid buy even if you end your hand by discarding 1 last card with Tactician. Team those two with some +2 action cards. Perhaps preferably Native Village for a more or less controlled hand size, and you got yourself a winner!
In these setups i tend to go for at least 2 Tacticians, 4 or more vaults, and as many Native Villages as possible. Might be a good idea to bring along a card with +buy if possible as well.
As you can probably can imagine, this also works very well as a defense system against attackers that try to clog up your deck.
Remember to not discard all cards, save 1 for the Tactician to work 🙂
A great combo if you can say so is vault and grand market.
For example I start with 5$ and buy a vault. Then I can buy a grand market in turn 3 or 4 (discarding coppers for 1$) which is a grand advantage.