Is it forbidden for a woman to go out for charitable work if that becomes widespread in society, and if not forbidden, how does that align with the decision to stay at home?

The charitable acts performed by a Muslim woman seeking the countenance of Allah are righteous deeds for which she will be rewarded. There is no objection to her undertaking them, provided she adheres to the hijab and modesty, avoids mixing with men, and works with other women to close the door to fitna (temptation). Muslim narrated from Jabir ibn Abdullah (may Allah be pleased with him) that the Prophet (peace be upon him) said to his aunt, "(Yes, so prune your date palms; for perhaps you will give charity or do a good deed)." If cooperation between men and women is necessary in this field, it should be through the mahrams (unmarriageable relatives) of the men. However, a woman's going out and personally engaging in charitable acts, traveling, meeting with men, and attending their gatherings is impermissible due to the resultant fitna and mixing. The default is for a woman to remain in her home, and she can perform many good deeds while at home through her husband, sons, or brothers.