Yes, we can migrate email accounts. If your current host uses cPanel we can take an automated backup and restore that as is. Email account passwords will remain the same and we do not need to know them.
If you use a proprietary control panel we can migrate emails using an IMAP sync. It is similar to creating two mailboxes in a mail client and dragging them (but automatic). For this method we will need to know a list of email accounts and passwords. Alternatively, we also give you access to do these mail migrations yourself.