Yes, this is what people do. Maybe suggest coffee rather than lunch (or coffee or lunch, and let her choose), to keep the time commitment on her part shorter. Make it first about asking for ideas in her field, rather than specifically about her company, so she does not feel put on the spot that you want a job with her. And assuming it goes well, hopefully she will give you some names of other people to talk to, and it will grow from there. Good luck!