Hey Greg, back in the day I used Google's inbuilt payments system but it was very clunky and they discontinued it.

In your situation, you can issue a licence key via Gumroad and either ask the user to plug it in or you can drop an activate button on the gumroad checkout page to help avoid the user copy/pasting incorrectly.

Either that, or you simply put the Chrome Extension as unlisted on the chrome store and just share the link with the user after checkout.