No, if the server is installed locally it's not needed an Internet connection. The computer is not connected to any local network, isn't it?
Are you using some firewall or antivirus software ?
I just attached a couple of reg files to enable the debug mode in the license server. Please, try this:
1. Double click the licserver_debuglevel (x64).reg file and accept the messages.
2. Restart the license server using the License Manager (Server->Stop Server/Start Server).
Now, you can get detailed information of the server using the "View Log" button. Use Forest and send us the log file when it fail.