Hi,
Yes this is possible. Stripe payment has a concept of creating a customer and then making a charge. Also, before making the customer, the credit card is verified and stored in their system, they generate a "token" on storing the card and that token is used for making the charge.
What this means is that if you have this token stored in your local database along with the customer, you can charge it again without needing the customer to enter their credit card details again.
This is a very simple modification and I can do this for you in couple of hours.
About me: 9 years of working experience in PHP and during this time I have built over 200 eCommerce and other solutions. I have worked on all commonly used platforms, frameworks, APIs and scripts. In stripe, I have used their APIs several times and I have even built split payment system using their API despite the fact that they don't have a native support for this.
Lets get started. I can start and finish this the same day.
Regards,
Asad