Until yesterday, UploadZen's Web Uploader (and File Upload Web Part for SharePoint) required either the Flash or the Silverlight browser plugins at the client side to support selection of multiple files for document uploads.
In all modern browsers (except, at the time of this writing, Internet Explorer) this SharePoint Solution Package now utilizes HTML5 features for selecting multiple files if the browser supports them.
It will still fall back to look for Flash or Silverlight support if that's not the case. If a future Internet Explorer update gets HTML5 support for selecting multiple files, then this will be utilized by UploadZen automatically.
We have also reviewed UploadZen's "Windows Uploader" (its ClickOnce-based rich-client / native Windows app). It should now work for all users on any Windows version using Internet Explorer. Do note that their default security setup may still prevent the Windows Uploader from being launched from your SharePoint Site, so the Web Uploader mode remains the optimal default GUI choice for UploadZen. But if you're willing to put in the extra configuration efforts in your local Windows setups to utilize the added capabilities of our Windows Uploader (such as client-side automatic zipping of files being uploaded), you should now have once again no issues launching it.