I am not really sure myself as this is not a minor change but I will try my best to give you some idea. The concept is that when the "Add to Cart" button is clicked it will have to send the data to a particular script using JQuery (this is the server side script for the ajax call). You will have to add JQuery code that listens for the button click and does that. Look for the following function in the "eStore_misc_functions.php" file for the "Add to Cart" button's code:
get_button_code_for_element
When the response comes back from the server you will need to draw the cart. The current shopping cart's function is in the "eStore_cart.php" file. You can also add your own function in that file to draw the ajax based cart. If you can keep the changes modular (for example add different functions and files for the functionality) then I will be able to incorporate the changes in the plugin down the path.