In the Orders view you can see the customers' orders. On each line you can find a button labeled "Mark picked up".

By clicking this when customers pick up their orders you can easily keep track of who has picked up their orders and who has not.

Clicking or not clicking the button doesn't affect payments in any way. If you do mark orders as picked up, make sure that you mark the correct customer's order as picked up. Marking the wrong person's order as picked up may confuse the customer and make them think that their order has already been picked up.