Just a heads up, I ran into this issue with firefox and unsupported flash (11.2) on f20. I ended up using chrome for a while until I came across Pipelight, which I installed (since I needed silverlight for something anyways), but saw there was a flash plugin as well. Enabling the flash plugin gives firefox flash 11.5 and I was able to get into the vSphere web client.
Pipelight is basically a plugin that emulates a standard silveright or flash plugin, but then pipes it through a wine session running the application (silverlight flash etc) allowing you to be running the newer (windows) versions. Still testing but seems to be stable so far.