In my county, you can sign up to get SMS or email alerts for school closings (and urgent weather, traffic, amber alerts, and lots of other things coming next month, including requests from the police to look for suspects or missing people, etc...).
No app needed. An app would probably need to be county specific since there is no standard for how or where school closings are listed.
If you are interested in learning to program, this would be a fairly easy type of app top create for your own use.
Alan