There's a lot of variation in the way databases and library catalogs handle foreign languages and that will affect how Zotero performs.
Although Chinese and other Asian and non-Roman language characters will display in Cornell's catalog, the Chinese or other non-Roman characters may not get sent through the Browser Connector into Zotero and be incoporated into citations.
If you need the original foreign language characters in Zotero, so they are incorporated in your citations and bibliography, use Worldcat Local (an interface familiar from Borrow Direct or Interlibrary Loan) which handles non-Roman characters quite well. Worldcat Local can be used for books, ebooks, reports, articles, and other material.
Whether the citations came from Worldcat Local, or elsewhere you may need to make manual adjustments to citations. But in those rare cases you have to type the entire citation into Zotero yourself, it's just once.
Also check Zotero's forums for more tips, strategies, and plugins, such as Chinese language Zotero guide
Set the language in the item record
Citations in bibliographies and footnotes are only as good as the accuracy of information in the fields of the Zotero item. If you're working in multiple languages and your citations are incorrectly capitalized, try telling Zotero what language the item is in so that the citation style can apply punctuation and capitalization rules appropriately.