No, this is not possible. 
--

Vice-presidential election procedure is defined by the [12th amedment][1]

> **The Electors shall meet in their respective states, and vote by ballot
> for President and Vice-President,** one of whom, at least, shall not be
> an inhabitant of the same state with themselves; they shall name in
> their ballots the person voted for as President, and in distinct
> ballots the person voted for as Vice-President, and they shall make
> distinct lists of all persons voted for as President, and all persons
> voted for as Vice-President and of the number of votes for each, which
> lists they shall sign and certify, and transmit sealed to the seat of
> the government of the United States, directed to the President of the
> Senate.
> 
> The President of the Senate shall, in the presence of the Senate and
> House of Representatives, open all the certificates and the votes
> shall then be counted.
> 
> The person having the greatest Number of votes for President, shall be
> the President, if such number be a majority of the whole number of
> Electors appointed; and if no person have such majority, then from the
> persons having the highest numbers not exceeding three on the list of
> those voted for as President, the House of Representatives shall
> choose immediately, by ballot, the President. But in choosing the
> President, the votes shall be taken by states, the representation from
> each state having one vote; a quorum for this purpose shall consist of
> a member or members from two-thirds of the states, and a majority of
> all the states shall be necessary to a choice. And if the House of
> Representatives shall not choose a President whenever the right of
> choice shall devolve upon them, before the fourth day of March next
> following, then the Vice-President shall act as President, as in the
> case of the death or other constitutional disability of the
> President.[Note 1]
> 
> **The person having the greatest number of votes as Vice-President,
> shall be the Vice-President**, if such number be a majority of the whole
> number of Electors appointed, and if no person have a majority, then
> from the two highest numbers on the list, the Senate shall choose the
> Vice-President; a quorum for the purpose shall consist of two-thirds
> of the whole number of Senators, and a majority of the whole number
> shall be necessary to a choice. But no person constitutionally
> ineligible to the office of President shall be eligible to that of
> Vice-President of the United States.

Since Clinton got 0 electoral college votes for the vice-presidency (shocker, I know), she would not be eligable for the vice-presidency vote.

Interesting side tangent: If Evan McMullin had managed to win Utah, it was _theoretically_ possible for him to be president, but not for his VP pick Mindy Finn to become VP. If somehow McMullin became president then Mike Pence would likely become his VP.

  [1]: https://en.wikipedia.org/wiki/Twelfth_Amendment_to_the_United_States_Constitution