You could use the user_register
-hook, to replace the username with the mail-address right after the account has been created.
It could be something like this (untested) in your functions.php
:
add_action( 'user_register', 'wpse216787_mail_as_username', 10, 1 );
function wpse216787_mail_as_username( $user_id ) {
$email = get_the_author_meta( 'user_email', $user_id );
update_user_meta($user_id, 'user_login', $email );
}