Yes!  As long as you can provide us with a cpmove file or a "full website backup" from cPanel, we can usually take care of the rest.  You may need to ask your host to create a cpmove file for you (they will understand what this is).  Some web hosts block access to these features, presumably to make it harder for customers to migrate away.  And some hosts don't use cPanel at all.  If this is the case, ask our sales team before moving over.  Some sites (such as WordPress-based sites) are usually very easy to migrate, while others (such as custom-coded applications) are not.  We never charge for account migration, but if we feel that the move is too risky for us to undertake, we'll let you know.