No, I’m not plotting any assassinations. 
If a political party’s candidate dies or suffers some kind of trauma that rendered them unable to continue the campaign, would the nomination automatically go to the VP candidate or would the party have to start all over?
Related question: what would happen if a newly elected president died between election and inauguration?
In the case of Mel Carnahan, his name remained on the ballot and he won his Missouri seat in the US Senate posthumously. His widow, Jean Carnahan, was appointed in the same manner as if he had died after being elected. Mrs. Carnahan had announced before the election that she would accept the appointment, so voters knew they were electing her.
Procedurally, the thing unfolded pretty much according the applicable laws of succession if he had died after being elected. For different offices, those procedures may differ. In the case of the US Presidency, it is already established that the president is succeeded by the vice president, so if a presidential candidate died and then won the election posthumously, the VP running mate would become the president.
State candidates are covered by state laws. Presidential candidates are covered by both state and federal laws.
In the case of a president, you don’t vote for a person in the first place. You vote for electors. That’s a subtle but crucial difference. Yes, you see the candidate’s name on the ballot. That is locked down weeks or months before election day to give the state time to print ballots. Depending on when the candidate stops running, his or her name will probably be on the ballot with no way of removing it. With sufficient notice, a new name could be added.
But if not then the election goes to the Electoral College, where the electors vote. Presumably the party would name the person they want to see as President, whether that is the Vice Presidential nominee or someone else. Presumably the electors would vote for that person. I say presumably since it have never happened and the laws are actually quite vague about what exactly must happen. The electors could revolt or vote their consciences or give up and go home, although I stress that is *extremely *unlikely in a real world.
If the President dies after the Electoral College, then the Vice President would become President at inauguration and select a new Vice President.
In the US, in 2016, in the presidential race, there would be widespread jubilation and celebration.
Both major party candidates. If there were any validity to “mind power” nonsense, I’d say we should all start wishing with great fervor.
Moderator Note
Let’s keep the political pot-shots out of GQ, please.
A Republican elector in Georgia said that he wouldn’t vote for Donald Trump in the electoral college. I mention this as a data point that electors don’t (always?) have to vote for the candidate they’re theoretically pledged to.
The last “faithless elector” was a Texas Democrat who voted for Lloyd Bentsen instead of Michael Dukakis in 1988.
Apparently the GOP is considering a related question right now:
The dead guy won against John Ashcroft. (1998 - Ashcroft’s best years were obviously ahead of him…)
The Wikipedia article mentions this in the only case of a dead person being elected to the senate, but that on 3 occasions a dead person was elected to the House.
So yes, if it’s too late for a substitute (as determined by the state laws) then the name says on the ballot. The Wikipedia article also mentioned that when Carnahan died, Ashcroft respectfully stopped campaigning. (two weeks before the election) When the lieutenant governor (and so, governor Carnahan’s replacement) said that Carnahan’s widow would serve if his name won, Ashcroft resumed campaigning.
In this case, it was about 2 weeks before the election, so far too short a time for some sort of party get-together to find and nominate a replacement and have the ballots ready.
Rules of the Republican Party - https://s3.amazonaws.com/prod-static-ngop-pbl/docs/Rules_of_the_Republican+Party_FINAL_S14090314.pdf (PDF Warning)
Based on the use of the word “otherwise,” above, Trump could be fired by the RNC, too. 
There are state deadlines when it comes to removing/replacing names on the individual state ballots. Some deadlines have passed, meaning Trump will be on that state’s ballot regardless of whether Trump remains the nominee. What remains to be seen is whether electors are bound to a nominee by name, or the party’s selection of nominee.
The Rules of any political party can be challenged in court and any attempt to replace a candidate elected fairly would be thrown out.
Some states have laws requiring electors to vote for the party candidate, others do not. Again, these are untested, only when the situation arises would you see the Supreme Court make a definitive decision.
A political party is a private organization. The rules of the private organization can be challenged in court. Assuming an attempt to change a candidate would lose as a forgone conclusion is a bit of a stretch.
Bumped.
See the last paragraph especially.
What? How do you figure?
In 2000, D.C. elector Barbara Lett-Simmons abstained in protest of DC’s lack of a House seat. A Michigan elector defected, perhaps by mistake, in 2004, and a whole bunch of Democrats voted for Colin Powell in 2016 as a last-ditch attempt to stop Trump.
In 1872, Liberal Republican/Democratic nominee Horace Greeley (“Go west, Young man!”) died after the popular vote and before the meeting of the Electoral College.
It didn’t make a difference because the Republican incumbent Ulysses Grant was the clear winner. Greeley’s electors were released to vote for whomever they wanted that wasn’t dead. A few tried to vote for Greeley anyway but their votes were discarded.
*The Democrats didn’t nominate their own candidate that year, just signing on to the Liberal Republicans’ nominee.