I'm wanting to ask this girl out, however I have found zero opportunities to run into her. She actually works at the same company but at a building that I would never go to. So, I'm thinking about asking her out through e-mail. Does this seem appropriate or cheesy?

If you have never said anything to this girl... I don't think email would be the smart way to go. You said that she works in a part of the building that you never would go... how do you see her? Whenever you see her... just say excuse me.. then ask her out... start with like a happy hour after work or something... then if that goes good, then a date.

Have you ever met the girl in person?

I have casually said hi to her and asked her how her she was but nothing more than that. If my senses are correct she is interested in me as well.

I very rarely see her and would like to not wait until I see her again.

She is in a totally different building that I would never go to. And besides, she works in the back office that I have no access to.