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Vault 15 is the easiest way to learn the location of Vault 13. The first thing you'll need to do to gain access to the vault peacefully is talk with Rebecca, a woman standing outside her tent, who happens to have a missing daughter.
Once you're inside the vault, level 3 has the missing computer parts for Tandi in NCR, and also the computer with the NCR spy information that will allow you copy onto holodisk. See NCR quest #10 and quest #1 below. The first computer in the room to the west on level 3 will also reveal the location of Vault 13, your main objective here.
Quests & Opportunities
Here is a list of some of the quests and other opportunities in Vault 15, each with a brief synopsis containing at least one way to solve it. The remaining ways, and possibly other minor quests I will leave up to you.
1. Give Spy Holodisk to authority in NCR:
After completing quest #3 below, make your way down to level 3 of the vault, kill the Khans there and retrieve the spy holodisk from Darion's computer if you make the Science check. Now bring that back to Tandi and Gunther in NCR for nice xp and cash.
2. Kill Darion:
First complete quest #3, and then see Zeke in the building in the northeast of the entrance area. He'll give you this quest and the key card for the vault. Darion is on level 3 of the vault but you'll likely have to fight your way to him unless you've got adequate Intelligence and Speech skills. If you can, tell them you're Pat and they'll be convinced you're new and allow you to pass. You can find some Combat Armor on level 2 in a northwest room. Once on level 3, do not forget to find the computer parts in the central room in a locker and access Darion's computer (after he's likely dead) to retrieve the spy holodisk for Tandi and Gunther in NCR. Finally, be sure to check the first computer in the room to the southwest to reveal the location to Vault 13. When you're done wiping out the Khans report to Zeke and then see quest #4 below.
3. Rescue Chrissy:
Find Rebecca in the starting area standing near a tent. Agree to go with her into her tent so that you can discuss things. She'll let on about her daughter gone missing, and while in conversation, will be alerted to an eavesdropper. Head on up to the northeast exit grid that a woman named Dalia is guarding. She should allow you to pass if you ask her if she saw someone. On the next area, which happens to have the entrance to the vault, you'll talk with Phil who's guarding the building. You can either be diplomatic and pass your Speech check without violence, or simply fight him and Karla inside before you get to Chrissy, trapped in a room. Phil has a key that disarms and unlocks the doors. When you return Chrissy to Rebecca, she'll arrange for you to see Zeke in the northeast building. He'll give you quest #2 and the potential to solve #4.
4. Complete deal with NCR:
After taking care of the Khans and killing Darion in the Vault, head back to NCR and speak with Tandi (see quest #10 there). That will wrap things up with regards to Vault 15. Be sure to give Tandi and Gunther the spy holodisk you picked up on level 3. See quest #1 above.