I guess "urged him to 'come through'" means "urged him to overcome difficulites (to succeed)".
Cambridge Dictionary defines "come through" as "to succeed in a difficult situation". So Trump implies that Pence should be brave under pressure and get his job done. I am not sure.
Vice-President Mike Pence - who as president of the Senate is due to oversee the session and declare Mr Biden the winner - said he welcomed the move. ................. By US election law, Mr Pence's duty in the proceedings is purely administrative, but Mr Trump has urged his vice-president to "come through".