I think most (but definitely at least some) of the departments of education at the state level release average SAT scores by high school/district annually.

MA: https://www.bizjournals.com/boston/news/2018/02/15/here-are-...

TX: https://www.bizjournals.com/dallas/news/2018/02/26/these-tex...

NJ: http://www.nj.com/education/2018/01/the_50_nj_high_schools_w...

IL: http://www.chicagotribune.com/suburbs/ct-school-report-card-...

IN: https://www.doe.in.gov/sites/default/files/accountability/sa...

VA: http://www.nbc12.com/story/33261679/look-sat-scores-show-wid...

CA: http://data1.cde.ca.gov/dataquest/