"Can I transfer my Windows system over to another drive if I haven't got the install disk or licence number?"
No, is the simple answer. Although the chances are that you have got a valid license (you usually know if you haven't!). Check the case of your PC for a green sticker with the code on it.
Your PC manufacturer should be able to supply you with a restore disc. If they can't then Microsoft can, for a fee of around £15.
What make / model of PC is it?