As noted above, your site is accessible over www and non-www. A very possible way that the session could be invalidated is by the user going from one version to the other. For instance, the "SHOP-STARTSEITE" link at the top is hard-coded to non-www. So, if a user comes and logs in on the www site and navigates there, the session can go away. It shouldn't, but Joomla (and some servers) can not keep the session on different sub-domains.
I'd highly recommend setting up a redirect from www to non-www (or vice versa) to see if that helps. It's also better for SEO purposes.
Thanks,
Alex