I think it should be possible, but should be very obvious to the user when an app is doing this (and should require extra permission to be granted).

Consider that I might want to record an encounter with law enforcement on the sly, and also want to keep my phone locked to make it harder to tamper with if it's seized.

Regardless, an app like Firefox should not be doing this, or even offer it as an option. There are certainly legitimate use cases even for Firefox, but I think with a web browser there's too much possibility for abuse and security/privacy issues to make it worth it.