It's going to depend on the plugin that you're using and how it saves the files. Mostly you'll get a set of files and a SQL file from the database. With those you can upload the files to the new server, set up a new database, import the SQL file and then change the DB connection settings in the wp-config.php file. That should be all you need to do.
If they don't have their cPanel details, then they should request them from their hosting company. It's a very simple thing that people do every day, and that will make life a whole lot easier. The biggest reason that you'd want this is that if the new host also uses cPanel they can do an automated domain transfer that copies everything from one server to another. That includes all of the WordPress files and database, but also includes everything to do with the domain like email boxes, DNS settings and everything else that's set up. All you need is the cPanel details.