Is it possible to have users register without having a password?

If you’re just wanting to add them to a mailing list, you really don’t need to create a user account for them. Just use something like this: http://wordpress.org/extend/plugins/mail-list/ or Gravity Forms to capture their email addresses.

The fact that you then say “to allow the user to buy on this website” seems to suggest that the user will need to log in – how will you authenticate them otherwise?

Hata!: SQLSTATE[HY000] [1045] Access denied for user 'divattrend_liink'@'localhost' (using password: YES)