An invoice is a standalone record - the end result - it cannot be used to generate anything new (except an income record when you mark it as paid, or a credit note if you refund it).
An order is like a template for generating invoices. If an order is set to auto-renew, it will generate an invoice each payment period (monthly, yearly, whatever) - as long as an administrator or a cron job fires the invoice generation process regularly.
To get a client to set up a recurring payment schedule, you have to email them the order renewal link that appears at the bottom of the order editor. If the client pays an individual invoice, it will just be that one invoice that gets paid.
worst case scenerio create individual forms for each client that goes on recurring plan and provide link to that form.
I suppose you could, but it would be easier to create an order record and provide the renewal link rather than a link to a form! A form is meant to be used for self-service ordering of pre-defined products.