Try turning off the Firewall in System Preference - Security / Firewall.
And try here > Mac App Store: Sign in sheet does not appear.
Third party software such as Little Snitch will prevent the MAS from launching.
Launch iTunes. From the menu bar click iTunes / Preferences then select the Advanced tab.
Click: Reset warnings and Reset cache.
Quit iTunes. Try the App Store.
Well, still no luck..
When I try to sign in the text is visible, but nothing will happen after clicking on sign in, just spinnig gear!
Firewall is off on my system and I have no 3rd party network related apps like little snitch, etc...
It can access the Internet because when I click on "View in Mac App Store" for any app in my web browser, MAS will show up and works like a web browser, nothing more! Even it shows View in Mac App Store too!