The first thing to do when you find yourself attracted to a shy man is to question your feelings for him.
The reason for this is that shy men are more concerned with avoiding the social catastrophe of asking you out and failing than they are of getting you.
This is what prevents them from taking the initiative in the first place.
If a girl doesn't approach or initiate contact with a shy man, nothing will happen.
This is true, but handling this situation isn't as simple as you might think...
Once this happens, you will need to gradually assume a less assertive (more traditionally female) role, because he will then be more liable to get bored with you, wonder if he could do better, consider his other options (the perception of which his newly-found confidence will likely inflate), fool around, etc.