Vaults

Vault under Little Lamplight: Yes, it's part of the story Quests. Once you complete the quest "Picking up the Trail" you'll be pointed towards Little Lamplight. At that point, you can get into Little Lamplight by passing a speech check, having the "Child at Heart" perk that will open up a speech option to let you in, or complete the quest "Rescue from Paradise" to be let in. Once you're in, you'll get the quest "Finding the Garden of Eden" where you can get to the vault under Little Lamplight (Vault 87) by either going through Murder Pass, which is full of Super Mutants, or you can go in the back way, which involves you hacking a terminal that is turned on by the Little Lamplight mechanic. You'll need a Science Skill of 50 I believe to hack it. For the most part, this vault is very uninteresting and worthless. The only loot really in there is the G.E.C.K., which is a story item. There may be a few weapons and maybe a nuka-cola quantum or two around, but the main reason for coming here other than the quest is to find Fawks.

Vault 92: It's west of Old Olney, which is in the North Eastern part of the map. (There will be a lighter green square shaped area. On the right side is Old Olney, infested by Deathclaws but is close to a Nuka-Cola Quantum stash and has the Prototype Medic Armor in the sewer area) and the left half of the square will have Vault 92 in it.

Vault-Tec HQ: Not really. It will put the locations of Vaults 87, 92, 101, 106, and 108 on your Pip-Boy 3000's map if you haven't found them already, but other than that, you might find a few skill books and a nuka-cola quantum or two, but that's about it.

Hope this helps.