Their actual username? No they can't but they can edit just about everything else. Is there a particular reason why you want to allow them to change their username?

 There are 2 options when your users are using an email as the username (that I see)

1 - Buy a login module that allows login by email.

2 - Write your own module to change the relevant tables in the database.

I advise #1, the cost if something went wrong with #2 is probably going to be more expensive than #1.

With #1 your users can update their email and use the updated email to login.