Unfortunately, you cannot download and use the full version of Zotero on a Chromebook.
It is however possible to use many of Zotero's features via Zotero.org.
Please see this guide for Chromebook related Zotero questions: https://gouldguides.carleton.edu/zotero/chromebook
To use Zotero with Microsoft Word you must have Microsoft Word installed on your machine. It won't work if you use Office 365 in the cloud.
For instructions about how to download Microsoft 365 (including Microsoft Word) please see installation instructions on this Baruch College Technology Center page: https://bctc.baruch.cuny.edu/students/office365/