I think I may have found it, The People's Choice by Jeff Greenfield, published in 1995.
On Election Day in 1996, The Republican presidential ticket of Foyle-Block wins enough states to receive 305 electoral Votes, compared to 233 electoral votes for the Democratic ticket of Mueller-Vincent. Since 270 electoral votes are needed to win the Presidency, the issue of who should be the next President is thought resolved.
However, President-elect Foyle dies only days after the election in a freak accident. The Republican party then promotes his running mate, Governor Theodore Block, to the top of the ticket. Governor Block quickly proves that he is a mental lightweight, his nickname being Terry Blockhead, leading some to question his fitness for the presidency.
One of those questioning Block's fitness to be president is Michigan elector Dorothy Ledger, one of those forgotten people lost in a party's political machine who seems unimportant—until the fate of an election hinges on her actions. Ledger publicly questions whether she is required to vote for Block, and in so doing sets off a political firestorm and a minor rebellion among some Republican electors. A series of missteps by Block further exacerbates the issue.
This review mentions the horse:
It was a contentious presidential election filled with backroom shenanigans, but finally the Republican candidate won. A few days after the election, the president-elect went to Wyoming to appear in a parade with children with disabilities. The unthinkable happens when the president-elect is thrown from the horse, and DIES. Now what?
The fate of the nation now is in the hands of the Electoral College. Everyone agrees that the former vice-presidential elect is barely functional and certainly not qualified to actually be president. What will happen when the Electoral College meets?
The Google search which got me there was
novel president-elect dies before swearing in. It wasn't until I wrote this question that I realized I'd been searching a president dying, not a president-elect.