Whether or not a woman can be appointed to such duties is disagreed upon. One opinion is that they can not, because the following texts negate guardianship of women:
الرجال قوامون على النساء
Men are in charge of women
— Quran 4:34
لن يفلح قوم ولوا أمرهم امرأة
Never will succeed such a nation as makes a woman their ruler
— Bukhari
The defect in the testimony of a woman (Quran 2:282) is also relevant, since testimony is a type of authority. When a single woman can not be a witness then logically she should also not be the judge.
Similarly women are taught to remain in their homes, and to observe hijab, and to avoid mingling with men, and even to avoid talking with them when there is risk of fitnah. And men are commanded to avoid looking at women. Hence it is not possible for women to perform such duties in public.
However if it is somehow done within the concessions of the law: For example the woman is an old lady for whom the rules of hijab are relaxed and she is in a situation where other requirements are fulfilled, then some hold that it is permissible. The evidence for this are traditions like that of Samra bint Nahik or of Shifa' bint Abdullah. Similarly in certain matters the testimony of a woman is acceptable, and testimony is a type of authority.
This opinion however is objectionable because:
These traditions are weak or their authenticity is disputable.
It is at best the madhab of a Sahabi, it is not known that the Prophet ﷺ approved of it.
There is no explicit mention of a market in this tradition, rather it is possible that this happened somewhere else.
There is no evidence in this tradition that she was officially appointed. One does not need to be appointed as a magistrate or inspector to enjoin good and forbid evil. Rather it can be that she did this when she went out for some personal need and then when she saw something wrong she corrected it.
Ref:
وأما الذكورة: فهي شرط أيضا عند غير الحنفية، فلا تولى المرأة القضاء ... وقال الحنفية: يجوز أن تكون المرأة قاضيا في الأموال أي في القضاء المدني؛ لأنه تجوز شهادتها في المعاملات، ويأثم المولي لها
As for being a male: This is a condition near the jurists other than the Hanafis, so a woman can not be made a judge ... And the Hanafis say that it is valid for a woman to be a judge over matters in which her testimony is accepted (i.e. other than hudood and punishments) although the one who appoints her will get sin
— Fiqh al-Islami wa Adillatuhu
