These two quests are paired due to their close proximity, although Hole in the Wall does not need to be completed immediately after Vault 81. One benefit of completing Hole in the Wall as soon as possible is that you can gain another companion.
To find the vault, head to the western side of The Commonwealth. The vault is immediately west of the Chestnut Hillock Reservoir, which looks like a small lake with a skyline view of Greater Boston.
This well-maintained, self-sufficient vault has survived because of a sensible bartering system it has with outsiders. To access the vault, you will have to adhere to this policy. When you try to open the vault, you'll have a conversation with Overseer McNamara, who will request that you trade three fusion cores. Once you have those cores, she'll give you free roaming privileges in the vault, and the Vault 81 quest will conclude. While you're here, speak with Doctor Forsythe and offer a blood sample at his request. You can find his clinic on the lower levels beyond the atrium.
To trigger Hole in the Wall, you'll have to leave Vault 81, walk several paces beyond the vault's cave entrance, then return to the vault.
Head all the way back to Doctor Forsythe. There's a tense situation involving a sick child vault dweller. They need your help to retrieve a cure from a closed-off section of the vault. After you accept this quest, follow Bobby DeLuca to the secret entrance to the old vault area.
This abandoned section is relatively linear. You just have to contend with a lot of Mole Rats and a lot of stairs and turns as you make your way to the waypoint. When you find a robot behind glass, speak to it. Tell her that you're here to relieve her of her duties. Overjoyed, she'll open the door to her room, and she'll give you the cure. Deliver the cure to Doctor Forsythe. After the ensuing conversation, the quest will conclude.