Let's say if all of the Supreme Court justices resigned and the seats are empty. Is the president required to appoint someone? What happens if he does nothing?
The Court requires at least 6 associate justices to constitute a a quorum to conduct business according to 28 USC § 1.
The Supreme Court of the United States shall consist of a Chief Justice of the United States and eight associate justices, any six of whom shall constitute a quorum.
Should the Supreme Court have more than 3 vacancies, it will not be able to conduct business.
The President is expected to nominate new justices and let the Senate confirm them. The law does not state what happens if the President doesn't do anything. However, as a last resort, Congress can impeach the President.
TL;DR: Nothing special happens
Almost all cases (except diplomatic immunity and cases between a state and US/other state) do NOT require Supreme Court to be the first recourse.
As such, any cases that are brought to Federal Judiciary will be tried by lower Federal judges (circuit courts I assume) and then the circuit court decision on the case will be the law of the land until (at some point in indeterminate future) SCOTUS is re-constituted and overturns one of those decisions.
As a side note, SCOTUS refuses to hear most cases anyway, in practice, so as far as most cases are concerned it's as if SCOTUS doesn't exist already.
An interesting legal question would be what happens when two circuit courts contradict each other (in which case SCOTUS is where that'd be supposed to be resolved).
However, taking this approach may lead to adverse political consequences for the President - such as loss of popularity; failure to re-elect down the road at the milder end of the spectrum; to higher end of the spectrum where such (in)action may be used as an excuse to remove the President before the election via either impeachment proceedings in Congress or the use of the powers granted in the 25th Amendment by the Executive branch.
While some presidents may feel that any Supreme Court, period, might be an obstacle or impediment to getting their way, no President, generally, is going to leave a vacancy unfilled if he has the ability to put forward a nominee.
If I am president, and leave the Court without a working minimum to function (see Panda's answer), then I can operate without that branch of government, but it's not like I get unfettered power to do as I wish, even with a cooperative Congress.
There are federal US District Courts, and US Courts of Appeals above that. Those judges can issue rulings and, without the Supreme Court to review them, their rulings would stand as law, until a functioning SCOTUS could eventually take them up. So there would still be a check on the Executive Branch from the judiciary.
If, somehow, they didn't make rulings, the policies of such a president would only be as effective as an Executive Order, moreso than if there was a SCOTUS in place.
The appointments made by any president helps to shape the court, ideologically, for several decades. If I have a particular viewpoint, putting in my appointments can move the court in the direction I wish to see (or prevents it from moving in the opposite direction, if I can't get a majority my way), put enough bodies that agree with me so another like-minded president can actually get a majority in place that aligns with my beliefs, or it can stunt attempts to move the court another way.
If I'm president for X number of years, and there's a wave of change in the next election cycle, my like-minded justices serve as an enduring bulwark against out of control foolishness of those who don't agree with my own irrefutable wisdom.
By the same token, if I leave vacancies, then the next president, who may be my polar opposite in ideology, can stack the court in his/her direction by suddenly making a bunch of appointments, almost certainly skewing the court heavily in that direction, for many years to come. First casualty would be all of my policies that never had to face SCOTUS review. Next up would be the next president's agenda, which would have much smoother sailing with a court personally hand-picked and stacked by that president.
Instead of some of my policies already being enshrined as precedent from being "blessed" by the courts, all of them would be up to be struck down (which is much tougher if they've already been set as standing precedent), which is why I referred to even laws as potentially being no more enduring than an Executive Order, under this scenario. My own (in)actions would have further cemented the opposition's viewpoint for generations to come. That's not a "legacy" that any president is interested in.