Complex sentence: The office of vice president remained vacant for the remainder of Pierce's term, as the Constitution then had no provision for filling the vacancy, making the Senate President pro tempore, initially David Atchison of Missouri, next in line to the presidency.
Split into simpler sentences: The office of vice president remained vacant for the remainder of Pierce's term, as the Constitution then had no provision for filling the vacancy. The extended vacancy meant during almost all of the Pierce Administration the Senate President pro tempore, initially David Atchison of Missouri, was next in line to the presidency.