In switzerland we have an very popular payment method called TWINT
https://www.twint.ch/en/business-customers/our-solutions/online-shop/?lang=en
I would like to implement into my web app. Whats the easiest way to do it?
https://www.twint.ch/en/faq/how-do-i-as-a-software-developer-receive-the-twint-api/
Their API is private, so maybe ask them how is best to integrate, or just using one of their providers.
If you want this to be the only payment service you want to integrate with it might be a good idea to reach out to them.
If you want to add additional payment methods, you should check their service providers and see if anything checks your boxes.
Option 2 is likely to cause fewer setup and maintenance headaches. 