The VoxForge Speech Submission applet is a signed Java applet. Unfortunately, OpenJDK's (free implementation of Sun's Java run-time environment) browser plugin, gcjwebplugin, does not yet support signed plugins.

From the Fedora Project Wiki:

Handling Java Applets

Upstream OpenJDK does not provide a plugin. The Fedora OpenJDK packages include an adaptation of gcjwebplugin, that runs untrusted applets safely in a Web browser. The plugin is packaged as java-1.6.0-openjdk-plugin.

• ...
• The gcjwebplugin adaptation does not support signed applets . Signed applets will run in untrusted mode. Experimental support for signed applets is present in the IcedTea repository, but it is not ready for deployment in Fedora.
• The gcjwebplugin security policy may be too restrictive. To enable restricted applets, run the firefox -g command in a terminal window to see what is being restricted, and then grant the restricted permission in the /usr/lib/jvm/java-1.6.0-openjdk-1.6.0.0/jre/lib/security/java.policy file.

Therefore, the Speech Submission app will not run with OpenJDK.  You will need to download a current version of Sun's Java run-time environment and link it to your browser using the instructions on Sun's site.