If a Senator happens to die while in ofice, what is the succession procedure?
You can find your answer in the 17th amendment to the U.S. Constitution.
Not only did the 17th amendment establish the process for electing senators, but it also explained how vacant seats are to be filled.
If a vacancy occurs due to death, resignation, or expulsion the amendment allows the state legislature to empower the governor to appoint a replacement. That replacement would hold the seat until the end of the senator's current term or until a special election could be held.
The only exception is Arizona. The Grand Canyon State requires a special election for all vacancies and does not allow any temporary replacements.
According to the U.S. Senate's official website, there have been 174 men and women appointed to fill vacant seats in the Senate. Of that total, 55 have subsequently been elected to the seat, 55 were defeated, and 64 chose not to run or were unable to run.
Here is a list of those appointments.
—The Fact Monster