Learn Turkish: Siz yeni müdür müsünüz?

The question particle follows 4-way vowel harmony. Since the last vowel in müdür is ü, the question particle becomes mü. Then, the "to be" suffix for siz also follows harmony, becoming sünüz. Put together, you get müsünüz.

Why does the "to be" ending attach to the question particle instead of "müdür"?

In Turkish yes/no questions, the personal suffixes (like "am", "are") always attach directly to the question particle (mı/mi/mu/mü). You leave the noun or adjective alone. So it is müdür müsünüz? and never müdürsünüz mü?

Since siz is plural, does this mean multiple people are managers?

Not necessarily! Siz means "you all", but it is also used as a polite, formal "you" for a single person. Because they are asking if someone is "the" new manager, it is most likely a formal question addressed to one person.

Why isn't there a word for "the" before "yeni müdür"?

Turkish does not have a separate word for "the". The context usually makes it clear whether you mean "a new manager" or "the new manager".