The ways in which the VP could influence the make up of the Supreme Court would be a President with an unprecedented level of incompetence left him in charge of "all domestic and foreign policy" (that's the offer Trump's people are rumoured to have made to prospective VP picks), or if there was a tie in the Senate between the yea's and nay's in the nomination process, and the VP had to break the tie.
Obama's nominee to the Supreme Court is Merrick Garland, who is a moderate. It was a tactical nomination intended to give the (majority Republican) Senate no justifiable reason to refuse to hold hearings to confirm the nomination. There is no legally or historically justifiable reason for their obstructionist refusal to hold those hearings, it's completely outrageous and historically unheard of. But if Obama had nominated a judge with a progressive pedigree, half of the electorate would likely have supported the obstruction in the hope that the new President would nominate someone more palatable to them. At that time it was still primary season so there was a chance of a Republican nominee other than Trump. With the Garland nomination, the Senate, which has a Constitutional duty to hold hearings on the President's nominee and confirm or reject him or her, has been clearly shown up as a failure in its capacity to carry out its duties, and there's some hope that the Democrats will make some gains in the Senate on the back of people's frustration with that.
The Justice who died and created the vacancy was Scalia, a pig of a man and strict Constitutional literalist who was as regressive as they come on all sorts of social issues including women's rights, so the conservatives are all beside themselves over his replacement, as he was a hero to them. There's a sense of entitlement to have someone who reflects his views, which is ridiculous.
The new President will not only fill Scalia's seat, but in all likelihood will nominate the replacement if the Notorious RBG (the brilliant Ruth Bader Ginsburg, who is 83 this year). Two other Justices are in their late 70s and 80s, so it's entirely possible that the next president will appoint four justices. Out of a total number of nine.
With Scalia gone, the court is split evenly between conservative leaning and progressive leaning justices. So the next President and the Senate are going to have an enormous influence for the next several decades on the kinds of issues the reach the Supreme Court - reproductive rights, civil rights, gun control, religious freedom (and freedom FROM religion), free speech, the legal definition of "woman"... all sorts.
Which is why is has to be HRC making those nominations... the alternative doesn't bear thinking about.