Once you create an integration with a specific cart, you'll be able to associate the lists in Marketing Cloud with the cart. And then, any order placed in the cart will automatically pull the contact information to the specified list.
The following is a screenshot of the New List page that highlights the 'Integration with Carts' section where your integrated carts will appear for you to be able to associate the list with.

To learn more about each of these cart integrations, see: