How did you access Yahoo Mail on your old laptop? If it was via your browser, and you were automatically signed into YM, then the password will be stored in your browser itself. It's very easy to access such data but we'd need to know which browser you were using in order to provide you with instructions. (Any decent computer technician should know how to do that anyway).
If you were using a dedicated email client on your old laptop (such as Outlook Express or Thunderbird) let us know so that we can try to help you.