The Vice President of the United States, while the President of the Senate, is not a Senator and therefore not a "Member of the Senate." Only members of the Senate may introduce bills for deliberation by the Senate.
Practically speaking, however, the Vice President will have absolutely no trouble finding a willing member of their political party to sponsor whatever they need.
The Vice-President's service as President of the Senate is, in fact, for the express purpose of ensuring that a sitting Senator need not be called to the role - and thereby denied a vote in most matters.
From the website of the Senate itself:
Vice presidents cannot vote in the Senate, except to break a tie, nor may they formally address the Senate, except with the senators' permission. Initially vice presidents appointed senators to standing committees, regulated access to the galleries and supervised the keeping of the Senate Journal, but these duties were later removed.
It's important to note that they may not even address the body without being given explicit permission to do so.