Are they complaining about Apple Pay or in-app purchases?
Apple Pay works like a normal card payment and any good payment processor should offer easy-to-use scripts to support this. It works on the web (on Apple devices) and will work seamlessly in a web view. There is no extra fee and you still retain full control of the customer relationship.
In-app-purchases is the one where the payment goes through Apple. It seems like they're not asking for that currently for your browser extension, but keep in mind they will likely do insist on it for the actual iOS app, so plan ahead.
My fear is that they are going to try to force us to implement in-app purchases for the iOS app to register for the service, which I don't like because I just don't.