I've done a bit of follow-up research on this question and offer the following notes, mostly involving pre-1937 instances of pizzazz (and related spellings).
The oldest of the 'Pizzazzes'
The earliest instance I've been able to find of pizzazz is, oddly enough, in the form of an invented proper name. From "At the Theaters," in the St. Paul [Minnesota] Globe (January 17, 1898):
It is a fair assumption that the majority of the people who assembled at the Grand last night expected to see somebody besides [Henry] Dixey. They looked for a vaudeville entertainment with Dixey as the star. Not a vaudevill-ain made his appearance. Two petite and shapely young women and two active and enthusiastic colored boys constituted Mr. Dixey's only visible assistants. One of the colored boys, whom Dixey has christened "Pizzazzes," borrowed the rings, watches and handkerchiefs from the audience, and the shapely young women in page costumes adorned the stage in the first part, and obediently and mysteriously vanished in mid-air upon subsequent occasions.
This occurrence of "Pizzazzes" is such an outlier and is so idiosyncratic that I doubt its relevance to modern-day pizzazz. It’s harder to reach the same conclusion with regard to the phrase "on the pizzazz [or pazazz]," from the 1900s and 1910s, however.
Sightings of 'on the pazazz'
In his answer, Hugo notes an instance of “on the pizzazz” from 1913 in the Mansfield [Ohio] News cited by WordOrigins.com. That instance is actually is actually from almost a year earlier, in an article on slang that merits reproduction of a more extensive excerpt. From "Nix on the Rough Stuff," in the Omaha [Nebraska] Daily Bee (December 29, 1912, citing as its source the Chicago Journal):
Nix on the Rough Stuff
Chicago's Clean Language League Throws the Kibosh Into Low-Brow Lingo
The Clean Language league of America, which is plum nuts about being dead set against slang, cuss words, risque stories, purple ragtime and wriggly cabaret shindigs—not because it cares a whoop, but because such things always sound like heck to strangers—held a wild-eyed jamboree in Chicago and cooked up plans for a grand hallelujah campaign to induce everybody to climb into the pure-words wagon and swear off on throwing the low-brow lingo. Quite a considerable bunch of language bugs took the splurge and the enthusiasm was all to the velvet.
According to the dope that was passed out today by one of the high moguls, Tommy Russell, the main doings tonight was to pick out a publicity gang which will have the job of throwing this line of bull into every state in the union, being particularly strong on the schools and colleges, and not passing up the educational hangouts for skirts. The sideshow of the movement ill be to go after the kind of music that you hear in the all-night dumps and at public hog-rassles. Brother Russell declared, bo, that his crowd had already framed it up with some of the big guys in the music world to put the kibosh on this line of junk, and that it was only a question of time before they would have such pieces as "When I Get You Alone Tonight" completely on the pizzazz.
This turns out not to be an isolated instance, although other early matches use a different spelling of the word (pazazz). One of these other matches is from Frank L. Chance (the Chance who played first base for the Chicago Cubs and figures in the old-time baseball saying, "Tinker to Evans to Chance"), The Bride and the Pennant (1910), as the star pitcher for the University of Chicago discusses with his roommate Tubby Bentley the repercussions of his having flunked some crucial exams at school:
"Tubs, I am not going to go back home and tell the pater I've flunked," he exclaimed. "I am not going to tell the mater her little boy is on the pazazz. I am not going to confront Lucile and tell her I fanned in the pinch. ..."
Here it seems that pazazz is not at all a good thing, though what it means precisely is not clear.
A second instance appears in the Lawrenceville [New Jersey] Literary Magazine: Centennial Alumni Number (1910) [combined snippets, date confirmed at Hathi Trust]:
I had a terrible time trying to decide how to spend the evening. There were two musical comedies and an Ibsen play at the theatres in the village and the Swiss Bell Ringers were at the Conference. I had seen the shows at the Hullfish and the Applegate, and the Ibsen play, over the post office, didn't have any chorus. So three of us decided to go to Conference and put the ding-ding exhibition on the pazazz.
The sense of on the pazazz in these instances seems to be something like "on the down and out" or "on hard times"; unfortunately, the phrase doesn’t seem to have had much staying power. The only other instance of that phrase that a Google Books search finds is from the San Francisco Daily Times, volume 17, page 26 (1908); but regrettably, no preview is available for that item.
The Library of Congress's Chronicling America newspaper database, however, finds several additional instances from the period 1908–1913. From "Red Ear of Corn for Him" in the New York Sun (May 10, 1908):
"Got a punk dose [rotten bit of luck] out in Seattle once," said a sheet writer. "I was on the pazazz so bad that time that the eats looked as unreal as circus posters to me."
From "Wooden Leg Swings in New Crime Wave: Store-Made Limbs Used as Offensive Weapons by Restive Cripples," in the [New York] Evening World (August 5, 1908):
Yesterday Mr. Charles Williams, whose home is under his sweatband and who has one regular leg and one proprietary leg, but whose appetite is just as good as any man's who has a full outfit of legs, got piqued because they wouldn't trust him for a measly little 2-cent stamp down at the Wall street branch post-office and took the prop. leg in hand and just naturally put the whole place on the scrambled pazazz.
From "Proclamation: Know All Men by These Presents" in the Spokane [Washington] Press (September 9, 1909):
WHEREAS, the people of Spokane are approaching that season of the year when wind whistles through the brush and the furnace begs for food; and,
WHEREAS, Hay lids and straw roofs [straw boater hats] are beginning to look like last year's birdsnests, not to say what is technically known among good dressers as being "on the pazazz"; and, ...
There is also one instance from this period where pazazz appears without the prefacing words "on the," though the story that uses it is so farcical and jargony that the word might as well be meaningless gibberish. From Irvin Cobb, "The Hotel Clerk Expatiates on the Chin Whisker," in the [Washington, D.C.] Evening Star (May 23, 1909):
"But what Cip got wasn't a whiff to what they handed poor Ab over in Turkey. He certainly had the allah pazazz passed to him.
The emergence of ‘pizazz’ in 1937
As I note in my original question, and as Andrew Leach corroborates in his answer, pizazz in the sense of "flash" or "wow factor” burst on the scene in 1937, through the combined efforts of the Harvard Lampoon and Harper’s Bazaar (more specifically, Diane Vreeland). Some reference works assert, however, that Vreeland didn’t say "pizazz" at all, but "bizzazz. " From Stephen Silverman, Dancing on the Ceiling: Stanley Donen and His Movies (1996) [combined snippets]:
During the period Leonard Gershe was still conceiving of his story as a stage musical, the role model for his magazine editor was Vivienne Segal, Gene Kelly's costar in Pal Joey, "because," said the writer, "I was crazy about Vivienne Segal, also her timing." As he began to research the world of fashion journalism, however, a young editor at Harper’s Bazaar, Didi Dixon, set him off on a different course. "She worked with the editor in chief, Diana Vreeland," said Gershe, "and Didi would feed me expressions and things that Vreeland said, words like 'bizzazz.' "
Bizzazz? As Gershe explained, "The word 'pizzazz' actually came from Funny Face. It was 'bizzazz' in the picture. I did not invent that word. Diana Vreeland invented it, and the movie popularized it. For want of a better word, she would look at a layout and say, 'No, no, no, it has no bizzazz.' And I used the word in the picture. Somebody a couple of years later made a mistake and spelled it 'pizzazz,' which is much better. The hardness of the p makes it more effective. Vreeland and I once talked about that, and she said, 'Yes, I wish I had said 'pizzazz.'"
Gershe’s account of the origin of pizzazz appears subsequently in Ty Burr, The 100 Greatest Movies of All Time (1999); in Harry Haun, The Cinematic Century: An Intimate Diary of America's Affair with the Movies (2000); in Marie Brenner, Great Dames: What I Learned from Older Women (2000), and Kay Thompson, Eloise: The Absolutely Essential 60th Anniversary Edition (2015).
The trouble with this account is that pizazz appears in the March 1937 issue of Harper’s Bazaar, whereas Funny Face (and Pal Joey) came out in 1957. And by 1939, other sources were using pizazz, as I noted in my original question. This is not to say that Gershe is wrong in claiming that Vreeland pronounced the word "bizzazz"—but the timing of the spelling mistake he talks about is all wrong in his account.
On top of that, we have the earlier sense of pazazz (discussed in this answer) from the period 1908–1913 to consider. I have found no evidence that pazazz in the phrase "on the pazazz," had any direct influence on the subsequent word pizzazz, but it certainly made a splash for a while in pre–World War I U.S. slang, and I wouldn’t be astonished if some memory of the older phrase helped shape the spelling of the new pre–World War II U.S. slang term.
After a probable false start in 1898, the first use of pizzazz (usually with the spelling pazazz) arose in the slang phrase "on the pazazz," which had a meaning meaning roughly equivalent to "on the down and out" or "on hard times" during its brief period of currency from 1908 through 1913. The pizazz of 1937 (which subsequently also appeared spelled as pizzazz and pazazz) is so different in meaning that I'm skeptical of there being any direct etymological connection between it and the earlier pazazz. Even so, the earlier term may well have influenced the popular rendering of the later pizzazz, especially if (as some sources argue) the 1937 word was originally pronounced "bizzazz."