Some MLS systems allow agents to use a mapping feature to specify the boundaries of a search. Your agent would get the school district boundaries (most have maps online), then draw those boundaries as an overlay on the MLS map, and conduct a search.
If particular Zip Codes are entirely within a school district area, the MLS can be searched by Zip Code.
MLS listings (some systems, at least) do have fields for elementary, intermediate, and high schools. Your agent can search by, for instance, "High School=Robinson." However, some agents don't fill in those fields. And some who do put in incorrect information.
MLS listings also can be searched by community. If you identify certain communities by name that are within a school district, the search can be for all homes within those communities.
If you're using one of the online services yourself, your best bet probably would be to search for "x miles" around a particular address (the school's address). That won't be perfect, but will be a start.
Hope that helps.