The entrance to this vault is accessible from Park Street Station. It's own map marker will become available after you successfully cleared the entire vault and rescued Nick Valentine during the quest "Unlikely Valentine".
In the Overseer’s room, you’ll find the Speech Bobblehead on the table. You can also find V114 Interview Holotapes (#03, #21, and #87).
While escorting Valentine, from the main hall, you’ll pass through the restrooms and bathrooms then go upstairs then reach a sleeping area. Before that doorway is a barricade of Vault-Tec containers. On a small container, you’ll find an issue of Astoundingly Awesome Tales.
The Advanced Sliding door near the Nursery has some caps stash, pre-war money, and a Master-locked Safe inside. In the Vault’s Park Street Station exit, you’ll find several Vault 114 jumpsuits. After exiting to the street area, the Vault’s location will be marked on the map.