学校が/は市の保健所に通報しました is possible and means mostly the same. As to why not just say 学校が/は, I think there are at least two things. 1) As you mentioned, 学校/'school' is inanimate. It's not impossible to let an inanimate thing be the subject of an action (like 通報する) , but there is a certain tendency to avoid that. 2) 学校が/は will make it sound like an external observer is speaking, like a journalist, not 'we'='school'. Again, I think this is merely a tendency, though.
Here is a similar example from https://www.env.go.jp/earth/ondanka/midstudy.html
This is from a government website. If it was 政府が, it would sound a bit more like someone other than the government is reporting it.
I don't think this is in the same category as 二人で. If you compare AとBは二人で通報した and AとBは学校で通報した, the latter can only be read as using 学校で as location. (In other words, the subject-like interpretation becomes impossible.) I think in a sense the 学校で in this question replaces a subject, but 二人で doesn't.