Unfortunately, this is not available on Shopify at the moment.
If you invoice a customer for a subscription product, Shopify will process this as a one time purchase.
We recommend not using the invoicing system for subscription products.
We offer the ability to sell memberships with Ongoing (for non physical products)
When setting up the product on Shopify , you can specify that it is a non-physical product, and this will make Shopify skip over the 'shipping' step on the checkout process.
One way you could do a workaround for invoicing is to simply email the customer a link to the product URL, and they would be able to checkout on their own.
For monthly subscriptions, Ongoing will create an order every month every time we charge the customer, which can be used for record keeping.