Short story..
Adobe ended flash support for Linux @ 11.2 and the recent vCenter 5.5 Web Client requires version 11.5 minimum… As of recent the only solution was to just use Chrome which for me was sad as I’m primarily a Firefox user. While looking for a silverlight plugin for Firefox I came across Pipelight, which just happens to have a flash plugin as well.
After enabling the flash plugin I fired up the vCenter Web Client and sure enough there it was. Still testing out how stable it is but if you do decide to try it I recommend installing the vCenter Firefox plugin both natively and in the wine instance.