Update in regards to season 2 (I'm avoiding spoilers as best as I can).

Functionally speaking, between MacLeish and the new VP, Secretary Moss acted as Kirkman's mentor/confidant and is often shown to support Kirkman the way a VP would. This meant that for Kirkman, nominating a new VP was a less pressing issue than it otherwise would have been.

Kirkman ends up nominating a new VP in S02E14 "In the dark". When announcing the nomination, he addresses the fact that he did not immediately nominate a new VP after MacLeish, because he took his time finding the right VP.

In the beginning of the episode, Kirkman mentions that it's not necessary to nominate a VP (link)

You're both focused on the politics here. I think you're missing the point. The point is we don't need to have a Vice President right now. The line of succession of the presidency's already in place: Speaker of the House, President of the Senate, both of whom have been elected to their positions. Both have earned the right to serve, unlike anyone I could simply appoint.

Tyler, Fillmore, Johnson (Andrew, not Lyndon) and Arthur all succeeded Presidents who died who while they were in office. Not a single one of them appointed a Vice President.

Kirkman then continues that since there is no requirement for it, it's better to wait until the right VP is found.

The first man that I appointed Vice President was involved in a massive conspiracy that almost brought our country down and cost me my life. We're gonna get this right. We're gonna wait until we find the perfect candidate.

Later in the episode, he does decide to nominate a VP. The relevant part of his address to the nation (link):

We are all too well aware of the tragic and treasonous circumstances that surrounded Vice President Peter MacLeish.
Those circumstances left me uncertain about how to fill that role. I wanted someone with clarity of thought, someone who had a sense of purpose, someone who would speak their mind and challenge me to become a better leader, someone who understood that there is a time when principled opposition needs to give way for the greater good.

And to be honest, I did not think that I would find that candidate. But I was wrong.
So it is with great pleasure and pride that I introduce to you my nominee for Vice President of the United States of America, [spoiler redacted].

NO.

Not that I can find. That post is currently vacant* which would put the Speaker of the House as next in line (IIRC).

^{*I confess this does seem like something they should be addressing}

Yes. The role went to his original chief of staff (towards season 3) Aaron Shore.

However, for a long time the role was vacant following the scandal of MacLeish