For me this issue seems to happen every time the base machine reboots.
I have not worked out a permanent solution but the following is a workaround to fix it until the next reboot.

Note: This was tested on VCenter Version 5.5.0 Build 2414847

The Error

The error printout you see when logging in through the VCenter web client is:

Failed to connect to VMware Lookup Service
https://[fe80::20c:29ff:fe2f:a475]:7444/lookupservice/sdk -SSL certificate verification failed.

The fix

SSH into your VCenter machine. Note: This is not the same IP as the ESXi machine.

Now view the ls_url.txt file:

$ cat /etc/vmware-sso/ls_url.txt
https://[fe80::20c:29ff:fe2f:a475]:7444/lookupservice/sdk

Just to be safe make a copy of it.

$ mv /etc/vmware-sso/ls_url.txt /etc/vmware-sso/ls_url.txt.bak

Now edit the file:

$ vim /etc/vmware-sso/ls_url.txt

to match the following:

https://your.server.ip.here:7444/lookupservice/sdk

Now reboot the VCenter machine

$ reboot

Wait for the machine to boot up.
You can use the following to see when the machine will let you ssh back in:

$ watch -n 2 nc -vz your.server.ip.here 22

You need to wait for:

Connection to your.server.ip.here 22 port [tcp/ssh] succeeded!

Now you need to wait for the web service to start up
You can use the following to let you know when it is up and running:

$ watch -n 2 nc -vz your.server.ip.here 9443

Now you should be able to login to the VCenter web client. A refresh of the VCenter web page will be needed if you had it open.

The following is recommended

Make a good copy of the file so you can just copy it back next time it happens:

$ cp /etc/vmware-sso/ls_url.txt /etc/vmware-sso/ls_url.txt.good

Next time the issue occurs

You can just run the following:

$ cp /etc/vmware-sso/ls_url.txt.good /etc/vmware-sso/ls_url.txt
$ reboot

And now make yourself a cup of coffee while everything boots up.

VCenter is not booting

The following should only be run if VCenter is actually not starting up correctly. Note it usually takes quite a few minutes to boot and start.

Start the VCenter web service.

$ /etc/init.d/vmware-vpxd status
vmware-vpxd is stopped
tomcat is stopped

If they both say stopped as above then you need to start the service:

$ /etc/init.d/vmware-vpxd start
vpxd is already starting up. Aborting the request.

If you see the above printout then you just need to wait for a little longer.
You can run the following to wait for it to start:

$ watch -n 2 /etc/init.d/vmware-vpxd status

When you see the following it is all booted up:

vmware-vpxd is running
tomcat is running