Does this affect all sites that use Java, or only one particular site?
What version of Java do you have? You can check on this test page:
Please note that some security vulnerabilities in Java have been reported in the media recently, but Oracle has not announced an official position on what users should do. If you do not currently have Java 7 installed, you might want to install Java 6.
When you say you are bypassing proxy, do you mean you are using a service like a private VPN? I don't know how that would affect your applet. If the bypass operates at the system level, all traffic should be captured. If it is implemented using a Firefox add-on, maybe that isn't the case?? Can you use an external proxy like Fiddler2 to see whether the Java plugin is communicating as you want/expect?
I have a similar problem - my Applet opens in all borwsers - IE 9, Opera, Chrome, only doesn't open in Firefox. It asks for a particular Java plug-in 1.6.0_12 and I have the latest version 6 patch installed - 184.108.40.206. Is there a way to make Firefox ignore the plug-in required and run the one it has available?