No, this is not possible.
Vice-presidential election procedure is defined by the 12th amedment
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
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
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.
To address Eques' comment -
Even if all the electors who can do a mass protest vote and vote Clinton, it would still not be legally possible for Clinton to become VP. According to FairVote, there are 29 states that have legal control over who their electors vote for. I ran the numbers and they have a grand total of 302 electors, over the 270 needed.