Whatever service you choose would need to have a REST API.
Apps Script can make requests to the API using:
UrlFetchApp.fetch(url)
So, the first thing to do is find out if the service you want to use has a REST API.
This is assuming that requests will be made from the server, (script files with the .gs extension)
I don't know if there are any client side SDK's available.
If the service had a client side JavaScript library, you could use it from the HTML <script> tag
in a sidebar or dialog box.
I use both PayPal or Stripe to accept customer payments, but it seems like you want something different.
With PayPal and Stripe you can use both client side code, and server side code.
First the payment authorization happens with the client side library and then you can make the actual transaction with server side code.
If the service has a JavaScript SDK, that's for client side use.
Even though Apps Script is run on JavaScript, I don't know of a way to use a JavaScript SDK from server side code.
So, is it possible? Yes, and it could be easy or difficult depending upon the circumstances.
Like what's available for open source code and how good it is.
I'm providing a general overview but maybe someone has already written code for this.
Maybe look for GitHub repos.