You'll have to test this with your list.
I'd expect that June-Aug is the best time because they are preparing for the next year, but since school year FY are July 1-June 30, you need to get in during Sept-Feb.
As for day of week and time, think about it and test it. Most teachers get to school between 6-8am and a principal or super may stay later than 3pm. Also try lunchtime.