There have been two presidents that have not sent a message to the Congress describing the State of the Union. William Henry Harrison (WHH or Harrison 9) gave his inaugural address on March 4, 1841 and by April 4 of the same year he was dead. The first to POTUS to die in office. Some attribute his death to not wearing an overcoat while delivering the longest inaugural address on one of the coldest days in the city of Washington. But his pneumonia didn't set in until 3 weeks after the swearing-in. Before the 20th Amendment, the message to Congress didn't take place until December.
The other president was also a 19th Century POTUS, James A. Garfield. The fourth to die in office and the second to die of an assassin's bullet. Garfield was sworn in on March 4, 1881, twenty years after Harrison 9. While WHH maintains the title of shortest presidency, Garfield isn't far behind. He was shot at a train station on his way to his college reunion. Charles Guiteau was a disgruntled office seeker, but was denied a job. Guiteau shot Garfield in the back on July 2, 1881. Guiteau proclaimed "Arthur is now president!" Such a proclamation kept Vice President Chester A. Arthur from looking like he was involved. But Garfield did not die until September 19th. The gun shot wounds did not kill the president, but rather the attempts by his doctors to save the president's life. Since the address to Congress wouldn't take place until December, Garfield never had a chance to deliver a message.