A way to glitch into the Vault is to get out a Grappling Hook (such as the Crossbow, Plunger, or Ice Pick) and spam-click the Vault. There's also another way to glitch inside, which is by using a stroller and let the baby inside the vault. Doing this will get you into a second chamber. This one has a large golden sphere, also with a lock. You cannot interact with this one. Nobody has glitched into the sphere yet, but you can stand on top of the Vault by spam-clicking a Grappling Hook. There are many Toys around the vault for sale.
A new update has a way to get into the vault. If you go to Burt, located between the red bridge and the hospital, you'll find an underground place underneath his fishing hook. Inside, you will be asked to look for the three parts of a key, which after you go in the Pet Store. One should be straight in the cove ahead. The next one should be in the stairway area, while the last one is on the top of the tallest tree. When you open the vault with this newly-assembled key, there will be a small crown inside, standing on a pedestal. When you interact with it, it reads, "Too small for my head, give me some time to think about it...". Once the countdown in front of the nursery ends, its purpose will be clear.
If you zoom out all the way above the vault you will find a tree and the words, "No peeking :) ".