Sawyer Tools is designed to avoid the need for invoices. The goal is for you to collect payment right away.
If you want to offer customers a way to pay for a class later or in installments, you can set up payment plans.
When a student who isn't already registered shows up for a class, you can use the +New Order button at the top right of the roster view to charge them and add their name to the roster.
If the class already passed and they still owe you money, you can use the Quick Charge button at the top right of their profile under Contacts. (You can use this Quick Charge button in any instance in which you need to manually charge a customer.)