Don't let a girl sit by herself at lunch. If she is shy, then you have to step up and do the talking. If it's there, which I'm guessing it is, she will open up to you soon enough. From the sounds of it, she may lack some confidence, and if that's the case, you must display confidence so she can draw some from you.

You're in a pretty good situation right now. You need to step up a bit and don't miss out on this opportunity. I wish I was in your situation because I love talking to shy people and getting them to open up. Good luck and tell me how it goes.