Shopping cart contents section is empty for anonymous users
This is almost always due to some sort of relationship in the View's configuration linking from the cart order's line items to some other entity the user doesn't have direct access to view. Rather than try to find the right permissions, the easiest thing to do assuming you have other filters (or contextual filters) on the View to ensure the customer doesn't see other data is to disable SQL rewriting under the advanced settings for the View.

On a hunch, though, you should also make sure anonymous users have access to view their own orders. We have special code in the cart module to ensure access control works in that situation, and it could be the entire order is inaccessible and not just some related entity.