Zotero Citation Management

1. You can install the Zotero Connector on a Chromebook and connect it to your Zotero library (online ... when you try the first time it will give you a link you can use to enable that). That would let you add bibliographic data to your Zotero library even without Zotero proper installed. That does not get you the ability to generate citations in Google Workspace. 
2. While you could not use Zotero to automatically create citations, you can manually generate a citation within Zotero and copy/paste that into your footnote.
3. While you could not use Zotero to automatically create a bibliography, you could put all your resources for a given paper into a collection in Zotero, and create a bibliography for that collection that you could then copy/paste into your document.

This isn't as automatic as using Zotero on a Windows, Mac, or Linux PC, but would still be time saving.

Paperpile looks to do what Zotero does, but to allow the same kind of clever integration with Google Workspace that Zotero has for Word/LibreOffice. Paperpile isn't free, but it might be what you need if you find yourself unable to use a Windows, Mac, or Linux PC.

You could also consider EasyBib Creator. This is comparable to Zoteros' online functionality.