Question

If the president leaves the office and the vice president becomes the president who nominates the new Vice President

Answer 1

The speaker of the house

Answer 2

According to CNN, "the line of succession falls to the speaker of the House, then to the Senate president pro tempore, then to Cabinet members."
Source: http://www.cnn.com/2013/08/27/us/succession-presidential-and-vice-presidential-fast-facts/