In Prisoner of Azkaban, Sirius Black anonymously sends Harry a Firebolt broom after the demise of Harry's Nimbus 2000 in the Whomping Willow. At the end of PoA, Sirius writes Harry a letter and explains that, yes, it was he who sent the Firebolt:
There is something I never got round to telling you during our brief meeting. It was I who sent you the Firebolt – Crookshanks took the order to the Owl Office for me. I used your name but told them to take the gold from Gringotts vault number seven hundred and eleven – my own. Please consider it as thirteen birthdays’ worth of presents from your godfather.
Prisoner of Azkaban - page 315 - Bloomsbury - chapter 22, Owl Post Again
Throughout the entire series it's made very obvious that breaching Gringotts' security was punishable by death. Gringotts had tons of security measures in place (and for the sake of this question, I'm not talking about the successful break-in the trio pulled off in Deathly Hallows).
Vault seven hundred and thirteen had no keyhole.
‘Stand back,’ said Griphook importantly. He stroked the door gently with one of his long fingers and it simply melted away. ‘If anyone but a Gringotts goblin tried that, they’d be sucked through the door and trapped in there,’ said Griphook.
‘How often do you check to see if anyone’s inside?’ Harry asked.
‘About once every ten years,’ said Griphook, with a rather nasty grin.
Philosopher's Stone - page 59 - Bloomsbury - chapter 5, Diagon Alley
Why did the Gringotts goblins act on Sirius's written direction that the money for the Firebolt should be taken out of his vault instead of Harry's? Sirius was the most wanted wizard in Britain at the time and it would seem that the goblins wouldn't have followed such suspicious directions so blithely when they received the request for funds from, presumably, Quality Quidditch Supplies for the broom. Why wasn't the Ministry notified that a request for funds from Sirius Black's vault had been made? How is it possible that Sirius was able to purchase Harry's broom via third party? Why did Gringotts pay out?
★ I'm looking for a canon-based answer (the Harry Potter novels and three supplemental books, any J.K. Rowling interviews with relevant information, or Pottermore). I do not prefer an answer from the HP Wikia/Wikipedia.