It's honestly insane that these sorts of technicalities are even possible to block the vote. "I called dictatorshipsies and you weren't in the parliament building when you clearly, overwhelmingly said 'no', so I guess no takesies backsies." There probably ought to be some sort of provision in Korean law going forward that if it isn't possible to enter the parliament building, they can hold the vote elsewhere.
Worth noting is that the unanimous vote includes members of the president's party (as far as I can see from skimming headlines. They're dropping fast...)
Yonhap news agency cited the military as saying activities by parliament and political parties would be banned, and that media and publishers would be under the control of the martial law command.
He's president, not prime minister. Removing him would require an impeachment, which usually has a higher barrier then a no confidence vote, though I'm not familiar with Korean government.