I work on the 25th floor of my building. Working on the 25th floor has brought to my attention how many people are afraid of elevators. Today I received a frantic phone call from a new client saying that she is afraid of elevators and is on the 23rd floor because she couldn't be on the elevator alone and everyone got off. She asked if someone could come get her. I was covering the front desk and asked one of my co-workers to go get her. This has happened a few times. One woman clung to my arm the whole time we were in the elevator. Nice to meet you too.
I've been stuck in our elevators more times than I can count. They breakdown all the time. Just last night the button to call an elevator wasn't working. I had to take the freight elevator down. I'm nice though and don't tell the elevator phobs this.
Apparently there is no official phobia name for a fear of elevators.