Popularised by the 1938 theatre musical Hellzapoppin’ ¹, and by the 1941 film Hellzapoppin’ ² based on the former, the one-word form hellzapoppin’ represents a colloquial pronunciation of the phrase hell’s a poppin’.

¹ The 1938 theatre musical Hellzapoppin’ was written by the U.S. comedians John Olsen (1892-1963) and Harold Johnson (1891-1962), with music and lyrics by the U.S. composer Sammy Fain (1902-1989) and the U.S. songwriter Charles Tobias (1898-1970).
² Produced for Universal Pictures, the 1941 film Hellzapoppin’ was directed by Henry Codman Potter (1904-1977).

The phrase hell’s a poppin’, and its variants, mean events are unfolding in a chaotic manner; a state of confusion and disarray is taking hold.

In this phrase, the verb pop means to suddenly break open.—Cf. the synonymous phrase (all) hell breaks loose and variants.

These are, in chronological order, the earliest occurrences of the phrase hell’s a poppin’ and variants that I have found:

1-: From First Fam’lies in the Sierras (London: George Routledge and Sons, 1875), by the U.S. author Joaquin Miller (Cincinnatus Heine Miller – 1837-1913):

“It’s Sandy. Cut his foot, I tell yer. Blood clean up to his elbows. Blood all over the house. Bunker Hill all over blood. Hell’s a poppin’, I tell yer.”

2-: From a correspondence from Carbondale, Illinois, published in The Chicago Tribune (Chicago, Illinois) of Tuesday 1^st May 1877:

It was known that the two men, Hightower and Kendrick, had quarreled several times. […] On Saturday night one of the witnesses heard Hightower say: “Hell is a poppin’,” inferring that something serious would occur. Another one—of whom he borrowed a revolver—heard him say on Sunday morning that they (Kendrick and he) would shoot it out before 10 o’clock that day.

3-: From the Las Vegas Gazette (Las Vegas, New Mexico) of Saturday 13^th July 1878:

From Lincoln.
A private letter under date of July 5^th gives the following news from the seat of war:
Hell is popping around here! I am informed that three fights occurred on the Rio Ruidoso on the 1^st instant; one man killed, one wounded [&c.].

These are, in chronological order, the earliest occurrences of the one-word form hellzapoppin’ and variants that I have found:

1-: From the Battle Creek Republican (Battle Creek, Nebraska) of Friday 17^th April 1896:

Word came from O’Neill this morning that the Holt county vigilantees were on another rampage and hellsapoppin in that Godforsaken country. Men with firearms and long ropes are in O’Neill to wreak vengance [sic] upon some cattle thieves who were tried and acquitted by the courts. The vigilantees propose to make them take their medicine, anyhow, and the courts be ——.

2 & 3-: From the Santa Fé Daily New Mexican (Santa Fé, New Mexico):

2-: Of Monday 21^st September 1896:

SOCORRO COURTY [sic] REPUBLICANS.

There’s a case of “hellsapoppin” down in Socorro county. Windy Williams, Lawyers Kelly, Si. Alexander and Lloyd Freeborn intended fixing the primaries and going as delegates to the county convention instructed for the St. Louis platform. Sheriff Bursum, Collector Cooney, Estevan Baca and a score of other free silverites, who still think there is something left in the Republican party doctrines, differed with their goldbug friends, and during the melee Delegate Catron was soundly scored. The precinct convention in Socorro county didn’t do anything to these fellows. It elected as delegates to the county convention the following: Filomena Miera, H. O. Bursum, W. E. Martin, Estevan Baca, M. Cooney and N. P. Eaton, and it passed a resolution—not indorsing the St. Louis platform—but declaring for the free coinage of the American product of silver at the ratio of 16 to 1. This was immediately followed by the withdrawal of Elfego Baca, Judge Freeman’s law partner, for re-election as county clerk.

3-: Of Saturday 17^th October 1896:

SOCORRO REPUBLICAN SPLIT
Democrats Put Up a Strong Ticket and the Independent Republicans Indorse It.

As intimated in these columns yesterday “hellsapoppin” down in Socorro county. The Democrats met on Monday and nominated this excellent ticket:
[Follows a list of names.]
[…]
On Wednesday the Republicans of Socorro county went into convention and after a lively bout the Bursum faction triumphed. This ticket was nominated:
[Follows a list of names.]
Then the trouble began. Such life-long Republicans as Juan Jose Baca, J. W. Terry, E. W. Eaton and a score more of influential party leaders bolted the convention and on Thursday they held an independent Republican convention and designated a ticket of their own, indorsing many of the Democratic nominees, including Fergusson for congress.

4-: From The Muscatine Journal (Muscatine, Iowa) of Friday 14^th June 1901:

We are told the country is standing on the brink of ruin, that hope is dead, that McKinley ³ is an emperor, that the rights of the people are being subverted, that a plutocracy is putting its iron heel on the necks of the humble, that hellsapopping and only a few of the richest will survive.

³ The Republican statesman William McKinley (1843-1901) was the 25^th President of the United States, from Thursday 4^th March 1897 until his assassination on Saturday 14^th September 1901.

5-: From The Brockwayville Record (Brockwayville, Pennsylvania) of Friday 16^th October 1914:

Hellzapoppin in Brookville town again. The wheels of the chautauqua circuit have stopped in order to give Tom Hendricks another vicious throw at the Republican party in Jefferson county. This week’s Brookville Republican goes after the state and county ticket with knuckles on, and condemns everything and everybody that is connected with the Republican party. Fortunately the brainstorms of editor Hendricks have ceased to draw victims to the quicksands of The Republican’s flighty policies.