That's a pretty loosey-goosey question. Well, between two President's named Roosevelt and the Truman family, and if we count both vehicles and aircraft, you wouldn't believe how many accidents there have been involving Trumans and Roosevelts. Among other things, this includes Harry Truman running a stop sign in 1938 with his wife and daughter in the car and getting T-boned at an intersection in Hagerstown, Maryland and rolling his car 3 times.
However I think the incident that most fits what you are looking for was a famous accident involving Teddy Roosevelt who was traveling in a horse-drawn carriage with his personal secretary, the governor of Massachusetts and his Secret Service agent in Pittsfield, Mass. on September 3, 1902. The carriage was just leaving an event where Teddy had given an election speech. As they were leaving, the carriage was T-boned by a trolley loaded full of trolley company VIP's who were running late to get to the event to see the speech. The trolley driver had all his bosses on board the trolley, and they were admonishing him to drive the trolley faster because they were running late. Fearing for his job, the trolley driver was driving the trolley as aggressively as he could. The trolley came around a curve and the trolley driver found the Presidential carriage right in front of him, and unable to stop in time. The resulting accident killed one of the horses and Teddy's Secret Service Agent, and injured everyone else, all of whom were violently ejected from the carriage, including Teddy. Teddy got up off the ground and was furious and had to be restrained from physically attacking the trolley driver. The driver had to do 6 months in prison for manslaughter and pay a hefty fine, but the trolley company brass, to their credit, paid the fine on behalf of the driver and gave him his old job back immediately upon release from prison. Teddy didn't have anything to do with it, and if he had his way, I'm sure the man's future would have been quite different.