> How can I refer to the shopping cart using document.getElementById if it
> does not have an id?
You can't.
This is where jQuery is your friend:
$(".pane-uc-cart-0").val()
...will get the value of the FIRST element with the pane-uc-cart-0 class.
jQuery ships with Drupal, and is automatically added whenever there is
JavaScript, so you don't need to add the jQuery library using drupal_add_js().
FYI, class names in Drupal are usually added left-to-right, from least-specific
to most-specific. So, the last class is the one that is the most specific to
that particular element.
-Karl.