The admissions office or family life office equivalent should be able to facilitate an email list for the logistical purposes you described. For holiday cards though, your kid will probably need to do the legwork.
We had a big (huge!) Hollywood star move to our town to send her kids to my D’s school during covid. The school directory went private that year, but they did have it set up where people in each grade level could access other grade level parents’ email through the portal (but not directly if that makes sense).