So where on fucking EARTH did Sirius get the money for this?

As other folks have mentioned, Sirius did have his own account at Gringotts. And it seems possible (likely?) to me that Gringotts and the Ministry don't have much to do with one another, therefore his account probably wouldn't have been 'frozen' when he was incarcerated in Azkaban. Maybe there is some special magical password that a person can use to access his account if he doesn't have, or can't present his key in person?