What is the procedure for choosing a new candidate for the Republican party in a US presidential election in the event that the party's nominee withdraws or is otherwise removed from candidacy prior to voting day in November?
I don't know if the 2016 rules are different, but according to the RNC's 2012 rules, Rule 9 covers this case. (Emphasis mine)
RULE NO. 9 Filing Vacancies in Nominations
(a) The Republican National Committee is hereby authorized and empowered to fill any and all vacancies which may occur by reason of death, declination, or otherwise of the Republican candidate for President of the United States or the Republican candidate for Vice President of the United States, as nominated by the national convention, or the Republican National Committee may reconvene the national convention for the purpose of filling any such vacancies.
(b) In voting under this rule, the Republican National Committee members representing any state shall be entitled to cast the same number of votes as said state was entitled to cast at the national convention.
(c) In the event that the members of the Republican National Committee from any state shall not be in agreement in the casting of votes hereunder, the votes of such state shall be divided equally, including fractional votes, among the members of the Republican National Committee present or voting by proxy.
(d) No candidate shall be chosen to fill any such vacancy except upon receiving a majority of the votes entitled to be cast in the election
In other words, the RNC itself (rather than the delegates) chooses who would replace the candidate, unless the RNC chooses to have another convention.
For reference, here's the composition of the RNC:
RULE NO. 1 Organization of the Republican National Committee
(a) The Republican National Committee shall have the general management of the Republican Party, based upon the rules adopted by the Republican National Convention. The members of the Republican National Committee shall consist of one (1) national committeeman and one (1) national committeewoman from and the chairman of the state Republican Party of, each state.
(b) For the purposes of this rule and all other rules, "state" or "states" shall be taken to include American Samoa, the District of Columbia, Guam, Northern Mariana Islands, Puerto Rico, and the Virgin Islands, except in Rule No. 14, and unless the context in which the word "state" or "states" is used clearly makes such inclusion inappropriate.