I would most likely assume that the email was setup with POP. First thing you should do is check the account settings on the Vista machine.
If by some miracle they are on IMAP I would simply export the contacts from Windows Mail, check for a signature and copy that. Then setup the email as IMAP in Outlook on the new PC. All the email will download, then just import the contacts and setup their signature. Boom done.
If they are on POP there are a few ways of going about it. I've done this a few ways in the past going between Outlook Express, Windows Mail, ect.
IMO they should be converted to IMAP if at all possible. If you can convert them to IMAP you have two choices. Option 1, you setup their account as IMAP in Windows Mail on Vista, then drag their emails over and let them upload. Option 2, you migrate the emails into Outlook first then upload.
I would almost prefer to just copy the message store to the new PC. Then from there I can do all the work on the new PC. Install Windows Live Mail, import the emails into there. Then export them from Windows Live Mail into Outlook.
Reference:
https://www.howto-outlook.com/howto/migratefromoewlm.htm
I've done it many way. Some have even stayed on POP. These days I prefer to get everyone on IMAP and I try to get anything business related into Office 365 or Google Apps for email.
I've had to do some crazy migrations before going between various email clients. Sometimes had to use Thunderbird or eM Client in the process.
I think the worst one was getting Incredimail into Outlook. I was real happy I finally got that guy off of Incredimail. This was his business email sadly.