When I first read your question, I closed the tab thinking it was outside my area of expertise. However a few days ago, a fellow mod, /u/mimicofmodes shared an interesting article about how women are written out of history and it caused me to go back to my education history books, especially those that focus on New York City and look for Tammany Hall. I realized that if we're talking about Irish immigrants, politics, and Catholics in NYC in the 19th and 20th century, there will be talk of education. And when that happens, women will absolutely play a role.

I can't speak to daily operations at the Hall, but can confidently say women were a part of what was happening at Tammany Hall - and sometimes organizing in explicit opposition to actions leaders of the hall were taking. For white women in New York City, regardless of class or religion, power was in collective action in the form of social groups. The City was resplendent with reform groups, often working together but just as frequently working towards different goals. The Women's Municipal League was one such group that formed with the explicit goal of pushing back against Tammany Hall. As far as I can tell from newspaper articles and the group's few appearances in NYC school history, women in the group disagreed with Tammany Hall's position on school funding. Democratic leaders associated with the hall were pushing to expand the network of Catholic schools, while the members of the WML wanted them to focus more on the public schools.

A number of other women's groups crossed paths with the Democratic party and Tammany Hall operators and members through their reform activities. Some were advocating for public playgrounds and looking for donations. Others were part of the emerging hygiene movement and advocating for laws and policies that would benefit women and children. I tried to backwards engineer some women involvement with the Hall by looking for times educators crossed paths with politicians associated with Tammany, but that was a bit a fool's errands, because those men were everywhere!

That said, I hope someone who knows the history of suffragists in the city can provide you a better answer!