Send payment requests
This is possible with Craft Commerce 3 Pro.

Create an email that would trigger on an 'Invoice' order status that you configure. (or any other name you want)

You would make the order in the Control Panel, then mark it as complete (which will set the the default status), then change the status to 'invoice' to trigger the email.

In the email template, link to a front-end payment form with the order number appended as a param like this ?number=78nc4y83fn4830f4nu304 Then in the payment form, pass in the orderNumber param based on the url number param.

This will make payment on the order and mark it as paid.

You can then see all orders with the status of 'invoice' in the CP, and see the ones that are paid or unpaid. You could move the order to a 'completed' status once you verify payment and ship the order.

Hope that helps.