I have been wondering where the actual vote of the Electoral College actually occurs. Do they meet in the House/Senate chambers? Where is it actually done?
It doesn't. I believe the individual state ECs meet in their respective state capitals to cast their votes, which are then sent to Congress, which formally counts them on the January 3 after the election and declares either that someone has been elected, or can declare that nobody has been elected and a contingent election will be held (this has happened once under the current system, in 1824, although a contingent election was also held in 1800, that election would not have resulted in a contingent under the current rules).