[Quick Search] List no longer displayed

I have been using Quick Search on a website for a long while, but recently, my client called me and said that the nice ajax-ed pop up with the search results is no longer displayed.
(https://www.bluemonday-coaching.ch)

I have looked into this with Firebug, and strangely enough, the request for quick search’s “search.php” gives a 404 error (although the file definitively is there) – “GET search.php?s=test” results in 404. However, Firebug displays an answer (both source HTML and rendered HTML), so I am really lost in what is going on here…

I am no expert in neither jquery nor any other AJAX stuff, so maybe this is something really simple?