Have both the president and vice president of the US ever died while in office?

Yes. George Clinton died in 1812, Rufus King died in 1853 and Garett Hobart died in 1899.

Seven U.S. Vice Presidents to date have died in office:

1. George Clinton (April 20, 1812)
2. Elbridge Gerry (November 23, 1814)
3. William R. King (April 18, 1853)
4. Henry Wilson (November 22, 1875)
5. Thomas A. Hendricks (November 25, 1885)
6. Garret A. Hobart (November 21, 1899)
7. James S. Sherman (October 30, 1912)