Using Product Cart without cookies enabled.

On the storefront side ProductCart primarily uses server sessions, not cookies (for keeping track of the shopping cart contents). One exception to this is the Product Review feature, where ProductCart specifically asks the Admin whether they would like to use a cookie saved to the customer's computer to limit the amount of reviews that the same customer can post. You can specify how many reviews each customer can post for each product.

The "Recently Viewed Products" snippet in the storefront also uses a cookie on the customer's system to determine what products to show.

... but in terms of the actual contents of a customer's shopping cart, ProductCart uses a server session to keep track of that, so it might be possible for your client to use ProductCart in their environment, but we suggest that you turn off cookies and try to mimic the typical customer's experience as much as possible to see if it's going to be feasible for them.