This has little to do with Delphi, per se, and more to do with using Delphi to access your payment procssor's API. It probably belongs in another part of the site as it has nothing to do with VCL.
If you don't have any experience calling REST APIs in Delphi, then play around with some free ones first.
Then find a payment processor you like that has a public API and use Delphi to access it. They'll probably have examples of several languages, like js, python, php, etc. Not likely any Delphi example. So what? If it's a REST API, just use Delphi's REST debugger and go from there.
TMS Software has the VCL Cloud Pack which implements an API for PayPal.
https://www.tmssoftware.com/site/cloudpack.asp
They also have their FNC Cloud Pack (a cross-platform library) that supports PayPal and SumUp
https://www.tmssoftware.com/site/tmsfnccloudpack.asp
Stripe also has a REST API you could call with examples for several languages: Ruby, Python, PHP, Java, Node.js, Go, and .NET, as well as a basic Curl example.
https://stripe.com/docs/api
There's also a platform called Plaid that offers services to talk with several banking platforms:
https://plaid.com/docs/api/
(There are actually a bunch of services similar to Plaid you can use.)
But all of this is simply payment processing. If you're looking at using Delphi to build the next eBay, Amazon, or Shopify, that's a whole 'nuther can of worms, and I would not recommend Delphi as the implementation platform. That said, I've worked at a number of places that sold various services and did integrated billing -- some that supported healthcare providers like Doctors and Dentists, and others that just did various other specific services.
Explain your use-case and you might get some more substantive insights.