Most Helpful Guy
A lot of times, the approach will be based on the shyness of the guy doing the approaching, more than the shyness of the girl. By that I mean an outgoing guy will be able to approach a shy girl as easily as he would an outgoing girl, and a shy guy can still struggle to make an approach on a shy girl that he finds attractive.
Personally, if I am talking to a girl and I realize she is really shy, I work harder to maintain the conversation, to keep it going and to cover up awkward silences so she doesn't start feeling that "omg I should be saying something...". I generally try to steer the conversation more towards things about her as well, as a person is usually more comfortable in talking about themselves and things they know or do. I mean, I try to do that normally, because I find people really interesting in general, but I make a particular concerted effort if the person is shy. Growing up shy myself I realize how hard it can be, so I try to do the things that I know can make it easier.
Small talk can be awkward, as can too strong of a flirtation, so it's about finding those things that she is passionate about, and able to talk about easily. Fandoms, hobbies, pets... things like that. Maybe make probes about deeper subject matter, more philosophical questions that can spark back and forth discussion and not just a yes or no answer. Don't just ask about a favourite show and then make them move on to a different subject... dig down into the answer - if you don't know the show, ask them about it, and what they like about it. If you do, ask them what characters they like the best, why do they like them, do they know anybody like that in real life... get them talking, and keep them talking about things they are comfortable with. Most shy people WANT to be having a fun conversation, they just struggle with initiating or maintaining that conversation.
That should give you the idea, anway.