Chromebook users are currently not able to download the Zotero desktop app unless they run Linux concurrently with ChromeOS. For instructions on how to install Linux on a Chromebook, see this tutorial from Ubuntu.
Zotero on the web will still work, as long as the appropriate extension is installed on the browser.
To create citations from Zotero on the web, select the checkboxes next to the resources you want to cite, the select the Cite button.
A window will pop up asking you to select your citation style. Choose the one you need and your bibliography is created automatically. Copy and paste the bibliography into a Google Doc to save it.
Unfortunately, you are unable to create in-text citations from Zotero on the web.
You have to use Zotero Chrome browser extension to save citations to your Zotero account. Download the Zotero Chrome extension from the Zotero website.
To save a citation from your browser, select the icon displayed near the URL bar. This will work for articles in Library databases and anything on the internet, including websites, videos, PDFs, etc.
The icon will look different depending on the type of document you're looking at.