If you have good justification for why you need the data, you could probably get a quick answer by reaching out to the two major licensing companies: Music Theatre International and Concord Theatricals (the conglomerate that now owns Rodgers & Hammerstein, Samuel French, Tams-Witmark, and Andrew Lloyd Webber's Musical Company).
Based on some quick Google-searching, there could be 50,000+ organizations that produce theatre around the country:
–– 1,800 professional non-profit theatres in the US
–– 7,000 amateur theaters are paying members of the American Association of Community Theaters. So maybe double that for the total number of amateur theaters in the US?
–– 37,000 high schools (public/private) in the US. If 80% of those produce some type of royaltied production (non-public domain) each year, that's 30,000 schools.