The Lord Jesus and the disciples did not live under a representative republic form of government, but lived under a theocracy and the domination of a foreign power. However, I can answer that question: "Render unto Caesar..."
That means we are responsible to have to vote and I fully believe if the Lord Jesus had lived in a representative republic, He would have voted and so would the disciples, because He paid taxes just like the disciples. He "render(ed) unto Caesar..."
Good question though, and I am happy to be able to provide the Biblical answer.