Women respect and respond to confidence and humor more than most other things. At first, being nervous can be cute and somewhat endearing, depending on the woman. I'd recommend that next time you catch her looking at you, smile to her, walk strait up to her, look her in the eyes, and ask her to coffee or lunch.
If what you're saying is true about her looking at you a lot, she's probably too shy to say anything herself, so you could very well have that in common with her. Could make for good conversation over coffee.
I know it sounds cliche to say, just do it, but getting rejected isn't NEARLY as bad as the regret that comes from never saying anything. Even though you're nervous & your heart will be pounding, the courage she'll see in you from overcoming that fear is huge. Good luck!