Update to Firefox requires administrator privilege
I ran into this today updating to Firefox 61.0.2 and on restart up came a dialog saying "Firefox is trying to install a new helper tool" prompting me to enter my admin username &amp; password.  I'm generally not a fan of typing admin credentials into a dialog box for an application update for a plethora of reasons.

Solution for me was simply to download the latest release of Firefox, mount the disk image and drag it into my Applications folder.  This probably requires the logged in user to have administrative privileges, but at least does not require them to know any password - which was your stipulation.