You can use the store's "Batch Action" menu to print invoices (and packing slips) for multiple orders at once.
To do this, log into your Shift4Shop Online Store Manager and, using the left navigation menu:
Go to Orders > Manage Orders.
Look at the "New" tab (or whatever status the orders are currently in).
Place a mark in the checkbox for each order that you'd like to print.
After selecting your orders, scroll down to the bottom of the page and click on the Batch Action menu.
Once there, you will see an option to print invoices and an option to print packing slips for each order all at once.