The Michigan Republican Party has two factions, each with its own set of officers. Most of the delegates to the Republican national convention are determined by the state convention, not the presidential primary, so there may be two competing sets of delegates sent to the national convention in Milwaukee in July. However a court may also resolve the dispute. See this story.
The last time a major party had two factions, each with its own nominees, was in the 1930’s, when the Mississippi Republican Party had two factions, each of which put competing slates of presidential elector candidates on the general election ballot. The two factions were called the “Lily White Republican Party” and the “Black-and-Tans Republican Party.” The split had no effect on who won, because back then the Democratic Party was overwhelmingly dominant in the state.