Based on
Article 2, Section 2 of the Constitution, it appears that the President can appoint justices when he wants to.
It doesn't seem as if the President has a time limit in which he has to fill court vacancies. Considering that we have a lower-level court vacancy from
2005, I have my doubts that there is anything that could be done to increase the speed of the nomination process.
If, say, a Supreme Court vacancy came up in the year a President was up for re-election, and the President's party lacked control of the Senate, he could take a risk and wait to fill the vacancy until after the election in hopes of being reelected and having his party take control of the Senate.