What you could do is create your own cart instead of using the built in one. What I would do is add an ‘Add to Cart’ form button on the item details screen. You can set up the form to show the item and then get additional info, like the quantity and anything else you would need. Write the user’s email, the item, and any other related info to a cart sheet. In the sheet, also include a column to indicate if the order is fulfilled.
Create your own cart tab that will point to a placeholder sheet with just a column heading. Add an inline list to the tab that will show a list of items from your form response cart sheet. Filter it by signed in user and where the Fulfilled column is not true.
In the cart tab you could add a form button to submit the order. It wouldn’t contain any details of the items, but you could go through the sheet, once you get a new order record for a user, and mark all of the items as fulfilled.
This is a really simplified way to do this and it could get much more complicated depending on what your needs are, but it’s just an idea to get you started.