Where does all the money the Great Houses generate go? Like does it sit in vaults or do they invest it? I can imagine vaults for the Lannisters, but what about the other Great Houses? Where do they put it and what do they do with it?

Well, a lot of it gets spent to support their lavish lifestyles – the food and drink, the clothing, the servants and guards, the food and drink and clothing of the servants and guards, and so on, all of that has to be paid for, a lot of it in kind, but a decent amount in cash too. 

And a good amount has to be invested back into their estates – leaving aside how actively the Great Houses invest to increase the incomes from their estates, there are some not-inconsiderable bills for maintenance and repair of estates on everything from fixing the fences and hedges on your own lands so that the peasants’ animals don’t wander in to repairing the walls and roofs of your castle so that it can stand a siege. 

The balance would probably be held in strongboxes as cash and/or plate and other treasures in secure rooms, and would be held as savings against some future expense, like paying for a dowry or paying a ransom.