Does Zotero offer installable packages of Zotero for specific Linux distributions?

No. Zotero is distributed as a tarball that can be extracted and run manually or packaged by others according to distribution-specific standards. Some that have been packaged are

  1. debian/ubuntu/some chromebooks
  2. Snap packages, usable on multiple distributions: https://snapcraft.io/zotero-snap
  3. Flatpack packages, usable on multiple distributions: https://flathub.org/apps/details/org.zotero.Zotero
  4. debian
  5. ubuntu

If your distro is not listed above, check if a package is already available. If so, please list it here; if not, please check if a packaging request has already been made (and consider making one if not), follow your distribution's guidelines for requesting a package.