The issue seems to be the VSC service (Virtual Storage Console for VMware vSphere Server - NVPF) doesn't start after a reboot. It is set to [Automatic] but it attempts to start before the necessary vCenter services are running and gives up. This explains why it would work after installation but after a reboot it'd stop and throw the fault error. I manually started the VSC service then opened a new web client session and no errors when accessing the VSC summary page.
The fix I used to handle the reboot was to add a service dependency to the VSC service. I chose the vSphere Web Client service which is one of the last vCenter services to start.
Open a PowerShell session using "Run As Administrator" then enter the following command:
& sc.exe config NVPF depend= vspherewebclientsvc
(There is a <Space> after the equal sign)
I have rebuilt and rebooted the vCenter multiple times and it has worked every time. Tested with vSphere 6.0 Update 1b and VSC 6.1 on Windows 2012 R2, Windows 2012, and Windows 2008 R2.