A firmware password should not prevent you from using the Software Update pane to install a Big Sur update
But, even so, just enter the firmware password, if asked, and you will be fine.
The usual purpose for the firmware password is to prevent booting to another bootable drive (if you don't know that password.) If you don't have to boot from another drive (and just use the Software update pane), then you should not need to enter a firmware password.
If you don't own your Mac, then whoever supports your Mac might block software updates through a user profile.
And, you should try contacting whoever supports your Mac. Maybe you can get permission to do what you want....