Even down here in the deep south where we can grow winter crops the school calendar presents real challenges. For us, school garden success/fail is primarily a summer maintenance issue. If the garden can be in good shape when the fall semester opens we can carry through. If it’s been abandoned all summer and nobody has time to get to it for the first month of school - it’s pretty much game over.
You could check into the Captain Planet Foundation’s “Summer School Garden Support” program. CPF hires a team of summer interns to bridge the gap. I think it’s replicable on a school system-wide level. On a single school level, my daughter’s school signs up families to each take one week of garden maintenance and harvest and there is a task-master who stays in touch with them as they rotate through. That has worked really well. There is another local school here who auctions off the school garden plots at a spring fund-raiser event. The family that wins the plot gets to keep what they harvest but they have to maintain it until the kids come back.