Can a woman drink Zamzam water during her period? Why do many people sit down when they drink Zamzam water?

Yes, it is perfectly permissible for a woman to drink Zamzam water during her period. There is nothing to restrict this to any group of people. Anyone may drink it regardless of their condition or state.

It is always recommended to be seated when drinking, because it is the better position healthwise. However, it may be sometimes difficult to drink while seated, which means that there is nothing wrong with drinking while standing. This also applies to drinking Zamzam water. There is no special rule for it.
And God the Almighty knows best.