The Zotero Connector is primarily meant to work with the Zotero program on your computer. When people talk about using "Zotero", they're almost always talking about the Zotero program. While you can use the Connector to save to your web library, that's not the primary or recommended way of using Zotero, so just referring to the Chrome extension doesn't imply that you're using the website.
It recently became possible to install Linux programs on Chromebooks, so you can try installing the Linux version of Zotero:
https://www.zotero.org/download/
It recently became possible to install Linux programs on Chromebooks, so you can try installing the Linux version of Zotero:
https://www.androidcentral.com/how-install-linux-apps-your-chromebook