here]They were sued under the Sherman anti-trust act in 1935. [More here] more recently, in 1996 they picked a fight with the Girlscouts and lost. [More here.]
So it should come as no surprise that in their 97-year long history that there are some surprises. I was reading the book The Deejays by Arnold Passman and came across a doozie on page 28 "The society even went so far as to purchase it's own radio station as a strategic move." ASCAP owned a radio station says he. the adjacent text indicates it was around 1922. That was news to me. My first problem is that the canonical text, The Airwaves of New York does not mention this whatsoever. ASCAP appears in the index twice, both unrelated to the above. In short, I had doubts despite their history.
ASCAP first began collecting licensing fees from radio stations in 1923. This really makes a 1922 purchase unlikely. ASCAP was still defending the right in 1924 against the "Dill radio bill." The bill would have exempted broadcasters from paying royalties on copyrighted music. It was named for senator Clarence Dill, [D] WA. So while it makes sense in that tumultuous time they might have been driven to make such a purchase, the timing is off. But, the boyscouts owned an AM station in 1924... anything is possible.
But otherwise, there is nothing spurious about the claim on it's face. New York City was vibrant with radio in that era, and it was a little wild-west about the licensing and "the 1920s" is a big bracket of time. In that era it would have to have been a day-share AM station of some kind. Also note that they bought one, they didn't build one. So what I think we're looking for is a change of owner between 1922 and 1924 which had to have been very brief.
I went through a number of suspect records and quick changes in ownership and also short-lived stations and found nothing. I am left to pick which book I consider more authoritative: Airwaves of New York or The Deejays. I'm afraid I have to pick the former lacking any further evidence.