Unfortunately, the Zotero desktop application and browser connector are no longer included on the library computers' lab load, which means they cannot be used on those machines. This also means integration with Microsoft Word and Google Docs will not work.
If you already have a Zotero account and want to access materials you've previously saved, you can access your web library through the Zotero website.
In order to add new sources to your Zotero web library, you will either need to create records manually or upload PDFs and attempt to retrieve the item's metadata.
If you're able, downloading the Zotero desktop application and browser connector on your own device will allow you take full advantage of their features.