A client of mine has a big php website, and wants to add shopping cart functionality. Is it possible to use Code Igniter's shopping cart class, without writing the whole site in the Code Igniter framework?
How?
There is no point. The Cart library is just a basic wrapper for the session library. It won't get you anywhere near as far as you think.
Yes, you can. Just change all the dependency on CodeIgniter, especially the sessions and configs.
Although I'd recommend that you look into webforce cart instead of messing around with the CI Cart class.
Yeah should be fine, just build your main controller using the shopping cart class, so it's basically just a shopping app. After it's built match the design, and maybe the htaccess / routes to have the same url structure.
It all depends on the scale / bespoke nature of ecommerce you are planning. A possibly quicker option would be Magento http://www.magentocommerce.com/ which I can't speak highly enough of. It's definitely my first port of call for ecommerce when I'm doing development jobs.
Good luck.