It seems to me that that any attempt to sabotage a carriage would be iffy. While carriage accidents could certainly be deadly, it would be really hard to ensure a deadly accident, I think. You might be able to cause damage to the carriage, but chances are that your victim will just be stuck by the wayside for a few hours or get a bump. Because carriages don't go as fast as cars and are pulled by living animals, it's harder to have sudden, catastrophic failure due to one point of mechanical error. Not impossible, of course, but harder to make happen on purpose.
Speed is going to be a factor. Maybe someone accidentally-on-purpose spooks the horses to get them running, and someone else accidentally-on-purpose gets in the way so they come to a sudden stop? That might cause an accident.
Is there any way to sabotage the route? Maybe a bridge goes out, or there's a rock fall from a great height? A concealed hole in the road that might cause the horses to stumble?
I'm trying to think of the details, but I know William Seward was in a bad carriage accident in April 1865, and was so badly injured that he was in bed, half-dead, when the same group of assassins who killed Lincoln tried to kill, too. The assassin sent to kill Seward nearly beat his son to death and injured his daughter, but was thwarted in his attempt to kill Seward by the apparatus wiring Seward's mouth shut. Maybe you can find some more details on the accident itself if you google it.
ETA: It was a splint applied to Seward's broken jaw that saved his life, not an apparatus wiring his jaw shut.