The term has faded into disuse, like many other Broadway nicknames.
Wikipedia: O. Henry
William Sydney Porter (September 11, 1862 – June 5, 1910), known by his pen name O. Henry, was an American writer. O. Henry's short stories are known for their wit, wordplay, warm characterization, and surprise endings.
Wikipedia: Appian way
The Appian Way (Latin: Via Appia) was the most important ancient Roman road. Its importance is indicated by its common name, recorded by Statius (Sylvae, 2.2):
"Appia longarum teritur regina viarum"
"the Appian way is commonly said to be the queen of the long roads"
The Romans sensed the inherent nobility of the road imbued by the circumstances and method of its construction, and its utility to the Roman Republic. The via Appia was the paradigm of all subsequent Roman roads. It was the very symbol of the republic, which brought order to the terrain and peace and freedom to the peoples of Italy, at least in their ideals. Their greatest historian, Livy, took that point of view. He was not a Roman by birth.
23 June 1907, Chicago Daily Tribune, "Modern Rural Sports" by "O. Henry," pg. E4:
"I run over to New York every two weeks to see a show," says the farmer, hanging up the receiver. "I catch the eighteen hour flyer at Indianapolis, spend ten hours in the heyday of night on the Yappian way, and get home in time to see the chickens go to roost forty-eight hours later."
28 February 1928, Chicago Daily Tribune, pg. F4:
The Yappian Way.
Broadway is the street of broken hearts.
---N. J. R. in Life.
25 February 1937, Richmond (VA)
CONFRERE MORTON THOMPSON of the Hollywoods on Monday last spent an entire column to advertise his wounds...It all started when Reader's Digest credited Mark Kelly with defining Broadway as "The Hardened Artery"...Jack Lait, who has used the same phrase for several years, always put quotation marks around it...The marks meant that Mr. Lait was quoting some other writer or person...And so it isn't fair to accuse Jack Lait of cribbing it...Nordo we believe that Mark Kelly ever claimed the phrase as being original with him...Thompson testifies that he thought it up about four years ago -- applying it to Hollywood Boulevard...Then, disheartened when he saw it lifted, renamed the Boulevard -- The Yappian Way...Well, after getting a letter from a British actor the other day -- we don't feel very much like getting into these quarrels, anymore...We do not recall from whom we thefted "Hardened Artery" -- but it appeared here longer ago than four years...It may be found in our Vaudeville News files of 14 years ago -- credited probably to some actor...It is more likely we didn't credit it...Ditto for "The Yappian Way"...And so Mr. Thompson will simply have to think up another nickname for Hollywood Boulevard.
13 October 1943, Chicago Daily Tribune, pg. 18:
O. Henry nicknamed New York's Broadway as "the Yappian way," and the same pun has sprung, as an original flash of wit, from the lips of numerous illiterate wisecrackers.