The church scene in "The Public Enemy" is not a specific, named church. It's a generic, nondescript church that serves as a backdrop for the funeral of Matt Doyle, played by Cagney.
However, the film is set in Chicago, and the church likely was filmed in an actual church in the city.
It's important to note that the film was made during the early days of sound in movies, and many scenes were filmed in actual locations, not on sound stages.