You'll need to check the ShopSite docs to verify that this will work with your setup, and I have no idea how reliable this will be. My guess is ShopSite really doesn't want you to mess with the URL like I'm going to suggest.
The URL of a ShopSite cart has a parameter "sbid" ... sometimes this same value is for another URL.param -- "shopinfo". SBID is used by ShopSite to identify the cart, so in some situations you may be able to view the same cart in two different computers by passing the SBID from one to the other and constructing the right URL.
A simple test would be to put some items into a cart on computer A, then open a cart on computer B and replace B's SBID with A's SBID. Not sure how you discover A's SBID and pass it to B.
Graham