NEW YORK—Many Broadway productions are scrambling to resume ticket sales in the coming days to welcome theater-goers this fall after city and state leaders have green-lit a reopening of the Great White Way at full capacity by mid-September. “We remain cautiously optimistic about Broadway’s ability to resume performances this fall and are happy that fans can start buying tickets again,” Charlotte St. Martin, president of the Broadway League, said in a statement Wednesday. New York Gov. Andrew Cuomo said Broadway theaters can reopen Sept. 14 and will be allowed to decide their own entry requirements, like whether people must prove they’ve been vaccinated to attend a show. Selling tickets will allow theaters to gauge interest before stages open, said Robert Mujica, Cuomo’s budget director. Actors’ Equity Association, the national labor union representing more than 51,000 actors and stage managers in live theater, said the news meant the theater community is …