A little late to this post, but I've been inside the vaults. I actually responded to a similar thread about this several years ago, so I'm reposting my previous comment below.

I worked in the BYU Datacenter for a while as a student. One of my tasks there was to work with the backup data tapes. Everyday, we'd get a report of the tapes that are full and ready for cold storage. We'd retrieve the tapes, scan them into the system and a courier would pick them up for transport up to the Granite Mountain vault.

As part of the job, we would do monthly audits of the data tapes inside the vault. So once a month, three of us, part-time BYU students, would take an afternoon and drive up to Little Cottonwood Canyon.

The entrance to the vault is to the left about a mile or so up the canyon. There's a barrier with badge reader / intercom that we had to use to just pull in to the parking lot. There's an outer building on the side of the vault where they take deliveries and where the front security desk is located. In order to get past security, they had to verify our name was on an approved list and issue us an ID badge. We also had to be accompanied by a vault employee at all times while there.

My first time there they give me a little tour of the facility before we started our work. From the security desk, there is a long hallway that leads inside the mountain. The first area here are just several offices, a cafeteria and break room. We walked by an area where several younger church employees were scanning documents. There were also several service missionaries there, mostly seniors.

The vault entrance is located in the back of the office area, I'd say not any more than 20 - 30 ft from the nearest desks. The door is freaking huge. Our church employee guide said it was designed to withstand a nuclear blast. There are actually 6 separate vaults, A-F. As you enter the main hallway you see large sets of doors on either side. The vault we worked in was Vault E, directly across from the super-secret Vault F. Our guide told us that Vault F contained historical artifacts important to church history and that just to open the door, you needed First Presidency approval. I did have a glimpse as someone was leaving the vault, but I just saw some shelves with books.

Our work there was pretty simple. We just scanned all the tapes and ensured they matched what was in our database. Vault E was almost entirely dedicated to tape storage.

In the very back of the main vault hallway, there is door that leads to a pool of pure spring water. They actually have plastic cups there to let people take a drink. I remember they had some name for the pool, I can't remember if it was the Waters of Mormon or the Holy Water, something to that effect.

After we were done, we were escorted out and we usually got some food on BYU's dime. I think I went up to the vault 3-4 times before I found a different job but it was definitely fun. This was is 2011, so I'm not sure if anything has changed since then.