Heinlein's "The Menace from Earth" as nominated by JRE is probably the first science fiction story set in a moon base with an area of sufficient volume for humans to fly in by flapping artificial wings attached to their arms.
But it is possible that there was an earlier science fiction novel set in a moon base, or an earlier short story set in a moon base sometime in the 31 years between 1926 and 1957, that mentioned flying in vast indoor spaces in a moon base.
User 14111 in his answer to the question:
quotes from the novel The Man in the Moone by Bishop Francis Goodwin (1638):
The manner of our Travel to the Palace of Pylonas was more strange and incredible than any thing we have related, for at our first setting forth there were delivered to each of us two Feather Fans, like those our Ladies in Spain cool themselves with in Summer: You must understand, that the Globe of the Moon has likewise an attractive Power, yet so much weaker than the Earth, that if a Man do but spring upward with all his Strength, as Dancers do in shewing their Tricks, he will be able to mount fifty or sixty Foot high; and being then above all Attraction from the Moon's Earth, he falls down no more, but by the Help of these Fans, as with Wings, they convey themselves in the Air in a short Space, (though not quite so swift as Birds) whither they please. In two hours Time (as I could guess) by the Help of these Fans, we were carried through the Air those five Leagues, in all about sixty Persons.
This is probably the earliest science fiction story to mention using artificial wing like tools to fly on the Moon. And notice that the date is not 1938, 1838, or 1738, but 1638 - sixteen thirty eight.
But I don't think that it can count as being the first science based story about flapping wing flying on the Moon, since Bishop Goodwin describes flying outdoors in the open air and the Moon has no atmosphere.
But on the other hand, when did astronomers decide that the Moon could not have an atmosphere? Did many astronomers assume the Moon had an atmosphere in 1638?
I think that many astronomers believed the Moon had or might have an atmosphere even 200 years after 1638.
One astronomer claimed to have found formations resembling human structures on the Moon, and claimed they were artificial constructions, in the 1820s, and that presumably required belief in the possibility of an Lunar atmosphere. The corona seen in eclipses of the Sun by the Moon was proposed to be the Sun's atmosphere and not the Moon's atmosphere in 1724, and in 1806, but that was not proven until later in the 19th century.
So some people might possibly argue that Bishop Goodwin's The Man in the Moone (1638) was the first story in which the lesser Lunar gravity enable people to use artificial wings to fly on the Moon.
Others may argue that it was two scientific revolutions too early to be the proper first example. First, scientists had to discover that the Moon has no atmosphere, and second, space flight proponents and science fiction writers had to imagine moon bases with enclosed atmospheres.
Another example of a 19th century belief in the possibility of an atmosphere on the Moon is the Great Moon Hoax published in the New York Sun in 1835 about the discovery of life on the Moon, including intelligent man-bats who flew with their bat like wings.
Some people might nominate that as the earliest science fiction story about flying on the Moon, though it lakes the modern features of being done by humans using artificial wings inside a moon base with a contained atmosphere.
The Moon is hollow with an atmosphere inside in Edgar Rice Burroughs's The Moon Maid (1923). Since there is life inside the Moon, including intelligent beings who invade Earth, there are probably bird-like lifeforms flying around inside the hollow Moon.
If there are flying creatures inside the hollow Moon, some people might nominate The Moon Maid (1926) as the first story with flying in an inside atmosphere at the Moon. But others might object that the concept of a hollow Moon is scientifically impossible, and that probably none of the human characters strap on artificial wings and fly in the story.
Some sources suggest that the first description of a realistic moon base was in Brigands of the Moon (1930) by Ray Cummings.
If that is correct, the first description of flying indoors in a moon base, if earlier than "The Menace from Earth" (1957) would have to have been published in or after 1930.
The article in Analog about flying on the Moon inside a moon base, mentioned by Zeiss Ikon might mentioned whether the author thought they were the first to think of flying inside a moon base, or read about it in a story and did calculations to see if it was really feasible.