Versions of Zotero after 5.0 eliminated the Firefox in-browser Zotero application, so now all browsers (Safari, Firefox, Chrome, etc.) will work with Zotero via "Zotero Connecter" bridge extensions linked to a standalone Zotero program on your computer. See these Zotero blog posts:

You don't actually have to have Zotero installed or open to save items into your online Zotero library - you can just set up your Zotero Connector preferences in your browser so that you will be able to automatically send an item to Zotero.org via translator icons in the browser window (next time you sync those items will be added to your desktop Zotero library). You will not be able to save any item that does not have a translator icon, however.

An example: I don't have Zotero installed on my profile at the library reference desk, but I do have Zotero Connecters installed in both Firefox and Chrome, so if I see something I want to add to my Zotero library I generally can - and it will be added to my desktop library next time I open the Zotero application since I have auto-sync turned on.