Hi soldatenhund,
I took a look at your website and the reason you are experiencing issues with JLive! Chat is because you are using the old version component, plugin, and module. Sometimes our download servers cache the files and you may have downloaded the old version accidentally.
To fix the issue, please uninstall the JLive! Chat component, plugin, module and desktop application. Then install the new version by downloading it using this direct download link:
http://www.cmsfruit.com/component/jcommerce/?view=product&task=download&i=17 Using the direct download link above will ensure that you are downloading the latest version.
After installing the new version component, plugin, module and desktop application, then go to Extensions -> JLive! Chat -> Operators and re-download your operator access key file and then re-add it into your JLive! Chat desktop application. You should be all set after that!