eZ publish will not work properly without sessions.
If cookies are turned off you will get a new session on each page load, and since the basket is tied to the session it will be empty.
The only way to use eZ publish without cookies is to enable the <i>session.use_trans_sid</i> setting in <i>php.ini</i>.
However I won't recommend it since it is a security issue, see this page for more info. http://www.php.net/manual/en/ref.session.php