From which OS you're launching the vSphere Web Client ? If Windows 8.1 or Windows Server 2012 R2 with Internet Explorer 11, did you tried add the FQDN of your vCenter Server to Intranet Local zone ?
Check this another thread for possible workarounds: Re: vSphere Web Client integration plugin problems in Windows 8.1---
Senior Infrastructure Specialist
We are having trouble with the Client Integration Plugin 6.0 plug in. It simple wont run after being installed. Tried all three major browsers and none work. All browsers behave like it is not installed or they prompt to allow it to run but after allowing it...nothing. Ruled out and local app firewalls and the like. The Client Integration Plugin 5.5 plug in works fine when accessing our 5.5 environments.
It seems as if there is another security control that might be blocking the browsers from launching the app but we cant identify it.
Thoughts or suggestions?