Speak to Baby Burglar at X:253 Y:413 in Swan Lake Basin (enter through Eversun South if you cannot fly). He will ask you to buy a Broken Key from him for a certain amount of gold. The price changes each time you speak to him. The range is about 100-300 so buy the key when the price is closer to 100.
Visit Between Burglar in Eversun South at X:602 Y:590 near the mining area. He will ask you to buy a Key Mold from him for a certain amount of gold, which also changes each time you speak to him. The low end of this range is 200 gold.
Return to Swan Lake Basin and speak to Liam the Locksmith at X:630 Y:515 near the Guild Castle. He will charge a fee of 36 gold to make a Vault Key.
Once you have the chests items, speak to Bridget Love at X:604 Y:487 in Swan Lake Basin. Answer no to her two offers and receive the thief qualification after she reveals her identity. If you incorrectly answered yes, then she will run away, but will respawn if you walk far enough away from her.