The individual schools are a field in the mls, but not a required field. If the agent put in that optional info, it will appear when you look up property on the Redfin site or Estately.com. Not all sites carry that info from the download, but I know those two do..
If the house you are looking at does not have that info, click on the houses closest to it and one of the agents nearby may have included that data when they entered their listing. Then double check with the schools before making an offer based on schools.
If you know what school you are looking for, any agent can do a search by that school and send you the string of homes via email. Then you can target homes nearby that likely have the same schools,
Often the agent won't list the school because there are a few options for parents, and the school closest to the house may not be the designated school, and yet most of the children in the neighborhood attend it by permit. So if the agent can't guarantee the info is correct, best to leave it out altogether.
I can't find a public site with the ability to search by school info, so you will need an agent to do that kind of search. But if you use the map feature on Estately.com until you get a house that lists the schools you are looking for, you can probably figure out the ones with good potential pretty quickly.