I am trying to find out when grade schools, junior high and high
schools start their summer vacations in June 2006. Is there a
country-wide date for this? Or state-wide? Or what?

Dates can vary widely, even within counties. In my area, each town's
public school system sets its own start and end dates, and there are
dozens of public school systems in the county.

I don't think there would be an easy, one stop place to find this on
the web (or anywhere else, for that matter).

You might have to call and ask a couple hundred school districts to
arrive at a nationwide average.

Most public schools require 180 days of education a year.  Each school
district sets its own calendar.  They start in late August or early
September and end in late May or early June.  Various holidays and
vacation periods as well as emergency closing due to weather also
impact the schedule.