If "leaping laughter" were simply a form (leaping) that laughter may take, we would expect to see a multitude of instances of laughter leaping in various stories, novels, and poems—and so we do.
From Fanny Fales, "My Child," in Voices of the Heart (1853):
He frolics, and the old house rings,/ His silvery laughter leaping;/ 'Til with the day, he folds his wings,—/ My little Spring-bird's sleeping!
From "Our Chicago Letter" (July 18, 1872) in The Song Journal (August 1872):
Her marriage, which will have been consummated ere this letter is in print, brings to my mind many pleasant memories of her seasons in this city. I was, for the nonce, her guest, and sitting at her table, sipping claret, in drinking which she has charming abandon, the topic of her marriage, of which I had not heard before came up. With love or laughter leaping in her eyes, she drew from within her bosom a gold locket of elaborate design, and pendant upon a chain encircling her neck.
From W. Mowat, Dolerino the Painter (1882):
That meeting was not burdened with the sadness of parting, for Mr. Brown was as hearty an individual at table as could be found in the whole parish. His genial manner and fund of anecdote, as the night drove on, kept laughter leaping round the table, while the droll humour of Mr. Black was equally responsive to the parson's tale.
From "Three Books of Verse," in The Literary World (January 5, 1894):
Miss [Jane] Barlow's stresses are accurately placed; her cadences are musical; her matter is worthy to be sung; there are tears and laughter leaping from ambush after ambush. What wonder, then, that she has written a book that should be dear to thousands?
From W.B. Yeats "The Dedication to a Book of Stories Selected from the Irish Novelists," quoted in The New Ireland Review (December 1894):
Ah, Exiles wandering over many lands,/ My bell branch murmurs: the gay bells bring laughter,/ Leaping to shake a cobweb from the rafter;/ The sad bells bow the forehead on the hands.
From Aleister Crowley, Orpheus: A Lyrical Legend (1905):
AGAVE. Believe it or not!/ Here is true joy—the woodland revellings,/ The smile, the kiss, the laughter leaping up,/ And music inward, musings multiform,/ Manifold, multitudinous, involved/ Each in the deep bliss of the other's love;—
From John Ellis, Something Else: A Novel (1911):
Never again after tonight would these six sit down together—so much the better, perhaps. But, for the hour, how joyous they were! Joy was everywhere ; laughter leaped, like flame, from table to table : for was not this the last night of the Old Year and the first night of the New?
From Mary Hastings Bradley, "When a Man Loves," in Woman's Home Companion (February 1912):
"Perhaps I can wait till she returns," Carter murmured, a hint of laughter leaping to his eyes.
From Stuart Sherman, "The Humanism of George Meredith," in On Contemporary Literature (1917):
In Farina, The Shaving of Shagput, The Ordeal of Richard Feverel, Evan Harrington, and The Egoist one may possibly detect the influence of the comic spirit of Meredith's first father-in-law, Thomas Love Peacock, whose laughter leaped and flashed among the humbugs and intellectual follies of his day with much of the Meredithian lambency and gusto.
From James Joyce, Ulysses, (1922):
A coughball of laughter leaped from his throat dragging after it a rattling chain of phlegm. He turned back quickly, coughing, laughing, his lifted arms waving to the air.
From an unidentified poem in The Commonweal (1926):
We sat beside a hemlock blaze/ And felt the glow of coming days./ Ah, there were only just we two,/ And laughter leaping up the flue.
From Dorothy Parker, "Clothe the Naked," in Scribner's Magazine (January 1938), reprinted in The Collected Stories of Dorothy Parker (1942):
It was not the laughter he had known; it was not the laughter he had lived on. It was like great flails beating him flat, great prongs tearing his flesh from his bones. It was coming at him, to kill him. It drew slyly back and then it smashed against him. It swirled around and over him, and he could not breathe. He screamed and tried to run through it, and fell, and it licked over him, howling higher. His clothes unrolled, and his shoes flapped on his feet. Each time he could rise, he fell again. It was as if the street were perpendicular before him, and the laughter leaping at his back. He could not find the fence, he did not know which way he was turned. He lay screaming, in blood and dust and darkness.
In short, laughter has been leaping for more than 150 years in the pages of literature good and bad—sometimes flamelike, sometimes the merest spark, sometimes with gymnastic ease, sometimes with the ferocity of a wild beast, and at least once like stringy coughball of phlegm.
Given this history of written usage, I think it's fair to say that, figuratively, laughter may leap with as much legitimacy as sorrow may burst or irony drip. But the phrase does not inherently indicate anything more than leaping + laughter.