Since you installed it from the AUR, it was likely installed to /opt/firefox-bin, which is not writable by the user. If you want to use firefox's update checker instead of flapak (or maybe yay), you'll need to have write access to this directory or install it into your home dir (ex: $HOME/.local/share/firefox).
Note It's recommended to use the firefox from the software packages or flatpak. These versions have the update checker disabled.
Here's something should make it work for any user. Though it might break on an actual AUR update.
DISCLAIMER: This is probably very insecure. Only let people you trust update firefox. Seriously, just install from the official repository or use the flatpak.
Create a group named firefox and add yourself to it. Set the group owner of the firefox directory to firefox. If you have other users you want to be able to update firefox, add them to the firefox group as well.