I'm an independent extension developer who uses Stripe to process payments.
You can host the Stripe payment on your own website or use
https://extensionpay.com/ -- I've used both successfully. Both do require processing the payment either in a popup or tab, though.
I don't think there's an easy or policy-compliant way to process payments within extension pages. I don't remember if you can embed remote iframes in MV3 extensions, but if it's allowed, maybe you could use that approach (iframe to your own website with Stripe library). That said, it feels more complicated than integrating Stripe on your own website, using hosted Stripe checkout pages, or using ExtensionPay. The user experience is still acceptable if you use popups or tabs to process payments, which is quite common among extensions that accept payments.
If you want to move quickly, ExtensionPay is a great solution.
Regards,
Alesandro