I hear that many people won't allow a woman to be a mashgiach. Why is this?
The question revolves around of whether or not a woman may be appointed to a positions of power or lordship, known as Serarah in Hebrew. This is based off the Rambam's opinion (Melachim 1:5)
אין מעמידין אשה במלכות שנאמר עליך מלך ולא מלכה. וכן כל משימות שבישראל אין ממנים בהם אלא איש
“So too all leadership appointments…we do not appoint them (woman) except from the men”.
This is the Rambam’s position (as well as the Ritva's in Shevuos 30a).
Rav Moshe Feinstien in Iggeros Moshe (Y.D., II, sec. 44) was asked this very question ie. If a woman could be a mashgicha, and after discussing the issue and the opinion of the Rambam at length. He goes on to note that the Rambam is not the only view, and demonstrates that there are a whole series of Rishonim who disagree with the Rambam and are lenient. He says that in a dire situation where we are dealing with a woman’s livelihood, certainly these other opinions could be relied on so that she could continue to be a mashgicha for kashrus. In other words, he maintains that the Rambam opinion is lechatchila, and we would generally prefer to be stringent and rule like the Rambam. However, since this shaas hadchak, and there are major authorities- including, Ramban, Rashba, Ran, and Rabbeinu Tam – who disagree with the Rambam, we can rely on these other sources to give us the flexibility to allow this woman to be a mashgichas kashrus.
Personally, my mother worked as a mashgicha for the Star-k under Rav Moshe Heinemann for many years. Seemingly he held there was no issue. And to the best of my knowledge we were not in a dire situation (maybe my mom never told us :)
However, none of this can be taken as a halachic psak for any personal circumstance.
For further reading see here: http://text.rcarabbis.org/women-in-communal-leadership-positions-shul-presidents-by-aryeh-frimer/