I normally don't worry about any proof (like a signed agreement) for recurring billing. They pay one month in advance. If they don't pay, you just turn it off (after a few reminders).
But I've got a customer also ordering hardware this time. So this is preparation for a recurring charge that has not begun yet. There will be 5 monthly payments for the hardware then it becomes software-only.
I don't want to add a lot of friction, but what is acceptable for the Credit Card company?
I believe what you are looking for is a credit card authorization form.
It is unclear if you are being reactive to a CC company request or proactive to avoid an issue. Also, is the hardware recurring, or one time?