The Golden Vault is Act 7 of the first Chapter: Mafia Town.
Hat Kid has to collect 4 codes to the Golden Vault in order to unlock it and get the Time Piece. A clue to where the codes are will spiral out when Hat Kid touches the Golden Vault and makes a line on the path they travel to their location.
The Golden Tickets
"The Golden Vault" takes place in Mafia Town, which is a wide open location with many items to collect, and is identical to other acts in this chapter. As such, if you wish to know what collectables can be found here, please consult this section of the Mafia Town page. Check here for achievements. Since there's no threat in this mission, it's a good one to pick if you want to poke around looking for out-of-the-way items or Time Rifts.
After each mission Hat Kid will write about it in her diary, which can be found in her room under the swimming pool of pillows. The diary will only read the entry for the last act that was completed.
"Y’know, I’ve been seeing these tickets all around Mafia Town. Blue, Red, Green, Yellow – they’re kinda pretty! I HAVE been wondering where they all go though.
"And I’ve been wondering what the Mafia could be keeping inside those things too! Well, today’s the day I’m gonna find out!"