Is this a non-administrator account? If so, you may be better off installing Zotero in an Applications folder within your home folder instead of in the root Applications folder, so that you can update it without admin credentials.

Most people run Macs with admin accounts, so we don't test much with non-admin accounts.

You can also try manually reinstalling Zotero from the download page, and also deleting the 'updates' folder in ~/Library/Caches/Zotero (hold down Option when clicking the Go menu to access your local Library folder), but the problem might come back on the next update.