Hi Schin,
I can't help in terms of vector but you have two options... either pay someone who can vectorize it for you (there are people who do it for a nominal fee - just have to google).
Or you can try the technique I review in the Preparing your Calligraphy for Printing tutorial in which you do the minimum line of 1 px adjustment. Before doing that, be sure to use the burn tool over all the thin lines (not too much or it will cause a shadow behind it). Then make sure you select all the black lines (to speed the process, you can click on one section of calligraphy, then do Select Similar). Then make the lettering its own layer as described in the video, then do the minimum 1 px trick.
It sounds like a lot of work, but you should be able to get it done in under 20 minutes. The lines won't be as fine but will be sufficient for printing.
I can help you walk through it tomorrow if you get stuck.
