So show him that you want something more - ask repeatedly.
In general, you will need to carry the relationship up until the point that it is "official." When it has been established (or is otherwise clear) that the two of you are together, then you should start reducing your initiative and letting him take the reins.
So whatever you would normally do to indicate your interest to a confident man, double or triple it for a shy man (in proportion to his shyness): While I don't suggest that you overtly ask out a shy guy on an "official" date (this would too flagrantly undermine what he knows to be his role as described above), I do suggest that you suggest and then ask him to hang out under casual pretenses.
Of course, he will question this over and over in his mind, wondering whether you were asking him to hang out just to be friends, or if you wanted something more.
So, if you do choose to initiate, do so only during the early stages of dating, that is, until he builds his confidence and comfort around you.
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.
This can be humiliating, frustrating, even infuriating, and it can ultimately still end in failure.
(If nothing else, this will give you good insight into what it is like to be a man.) While all of this sounds like it might be too difficult to deal with - and many women will come to that conclusion - it is worth pointing out that if you can navigate through the difficult aspects of attracting and dating a shy man, he is likely to be completely faithful to you.
The first thing to do when you find yourself attracted to a shy man is to question your feelings for him.
The second caveat is that even a shy man has pride.
Though he knows his own limitations (as prescribed by his shyness), he also knows what a typical man is "supposed to do." So he recognizes, too, when you are doing it for him.
This will offend him if you flaunt it, or if it is evident due simply to the social magnitude of the task you've assumed (for example, if you take his hand and lead him through a crowd).
That offense will reflect negatively on you and the relationship.
If nothing else, this blog is an outlet for voicing my astonishment at the typical female's ignorance of the male mindset.