We frequently integrate Stripe into our FileMaker solutions for clients. It doesn't require plugins and works on all platforms (desktop, mobile, web). We've also used 360Works Plastic 2 which allows for numerous gateway options if the client requires it. It takes a bit more to setup and the plugin needs to be purchased, but again, it will allow for just about anything to work.
+1 on Stripe… It has a very nice API, and you can simply use the 'Insert from URL' script step with httpspost to interact with it.
Is there any functionality (i.e. ACH, recurring, EMV capabilities, etc.) that Stripe and 360 Works Plastic 2 do not offer that would make life easier?