President Kirkman, explaining why he doesn't need a vice president, refers to people who are in the line of succession to his office. He mentions the Speaker of the House and the President of the Senate. Technically, the vice president is the President of the Senate. President Kirkman is referring to the President pro tempore of the Senate, traditionally the most senior senator of the majority party.
At around 15 minutes into the episode, it shows Aaron Shore answer his phone which looks like a SAT phone, but in reality it was a radio. You can clearly see the bronze coloring on the back that connects onto a charger.