The short answer is: No, you can't, because you don't have to.
Seats is not a standalone application for selling tickets.
Instead, it is a tool that's meant to be integrated into a host system, typically an online ticketing system that also provides payment gateway integration, e-ticket generation, event management tools, and more.
This means you cannot integrate a payment gateway directly with Seats, because we don't provide ticketing functionalities.
If you're after a full ticketing system that includes reserved seating capabilities, we suggest taking a look at one of our customers, they're all awesome!