Can applets launch programs on the server? Yes, using CGI.
Any other implementation would be server dependent. Of course this requires a lot of coding and is non-trivial. A simple way to do this is not built into Java because that would require a special server. One of Java’s strengths is that it is web server independent. A number of organizations have developed special http servers that allow applets or other clients to run Java programs on the server in a secure environment. Most notably Sun’s Java Web Server implements a servlet interface for this purpose. The W3C’s JigSaw implements a similar idea called resource objects.