No need for a JSTOR account; past issues of the "Southwest Review", including the one that contains the article in question, can be downloaded for free from the Internet Archive:

https://archive.org/details/sim_southwest-review_1934-07_19_4

Quoting from the introduction by the editors, who transcribed Irving's handwritten manuscript:

"In 1832 Irving made his journey west to Missouri with Commissioner Ellsworth [...]. The plunder of this pilgrimage and of his reading was his three books on the Arkansas-Missouri lands, on the expedition to Astoria, and on the exploits of Captain Bonneville; but in his notes still survive other relics of this relatively unknown phase of his writing. [...] This sheaf of notes, 'Polly Holman's Wedding,' is a crude narrative of Kentucky, foraged, after Irving's immemorial custom, from some fellow traveler, and set down in the form of rough memoranda. [...] In many portions the manuscript is almost illegible, and errors have probably crept into this transcription, in which the editors have followed in so far as possible the exact orthography, punctuation, and spelling of Irving. [...] The original notes have the date '1805.'"

Quoting from the manuscript itself:

"During the night John Vertruce [?] who was considered a great singster called upon to sing— after much reluctance, he shut his eyes stuck out chin nose with great self satisfaction sang song Mother & daughters conversation about getting married.
    Oh daughter daughter daughter
    Oh hold your foolish tongue
    Im sure you cannot marry
    Because you are too young

Daughter
    Im 16 years & old
    and that you will allow
    I must and will get married
    For the fits come on me now—

    O daughter daughter daughter
    Where will you find a man
    O there is John the Plowman
    Will have me if he can

    He called me dearest [?] honey
    While milking of my cow
    I must & will get married
    For the fits come on me now."

Footnote by the editors: "Professor H. M. Belden, of the University of Missouri, states that this ballad still survives in Missouri."