How would I ask her this?

If it was me, I would invite her to the comfort of my own home and get to know her better there since you seem to have social anxiety. Your home can be a perfect place for you and her to have coffee and talk and get to know each other. The even better thing is that the coffee would be free since your home. That should eliminate the pressure of other people's presence quite well. I have the same problem, so I know what you're going through. You could even go to her house if you wanted.