Invoicing System?

Out of curiosity, are there any invoicing systems out there that allow you to automatically charge a client's credit card if you have permission?

For example, instead of sending an invoice, it would be more convenient to just charge their card automatically every day of that month instead.

I'm not sure if this is doable or if I'm just looking at the wrong features listed online.

You can use a subscription service like with PayPal subscriptions. Although, it will have to be the same price every month.

As for automatically charging the client a different rate each month, I'm not aware of any services that can do this.

Stripe