You need to check the following files on the old server:
Those contain the web users and groups, you need to copy those to the other server.
you can simply copy and paste the "web*" lines and paste them into /etc/passwd on the new server.
The same for /etc/group but then the "client*" lines.
Next step is to copy all the mailboxes and websites to the new server.
Tar the dirs and then copy the tarballs to the other server and extract them.
(I'm assuming you're logged in as root)
To tar the websites:
tar -pczf websites.tar.gz /var/www/
To tar the mailboxes:
tar -pczf mailboxes.tar.gz /var/vmail/
Copy them to your other server:
scp websites.tar.gz newserver:/
scp mailboxes.tar.gz newserver:/
log into your new server:
tar -zxfv websites.tar.gz
tar -zxfv mailboxes.tar.gz