Marjorie Kimmerle & Patricia Gibby, "A Word-List from Colorado," in Publication of the American Dialect Society (April 1949) has this entry for the term hitch:
hitch: n. A crick ; a limp. Used only in the expression "He's got a hitch in his git-along." Said of horses and people. OED, A limp, a hobble, an interference in a horse's pace.
As this entry suggests, "a hitch in [one's] git-along" (variously punctuated and spelled) appears to be much older and far more common than "a hitch in [one's] giddy-up" and its variants.
A number of instances of the "git-along" family from the first half if the twentieth century show up in database searches, starting as early as 1902. Here are the earliest ten that I found, ranging in year of publication from 1902 to 1939.
From "Along the Kansas Nile," in the Wichita [Kansas] Daily Eagle (July 22, 1902):
When a brakeman is lame, his comrades cay that he has "a hitch in his gitalong."
From an unidentified item in [New York] Puck (November 25, 1903) [text not shown in snippet window]:
"The pay is twenty dollars a month, and board," said pessimistic old Farmer Bentover, addressing the applicant for work, who had a pronounced drawl in his speech and an unpromising hitch in his get-along. "And I've got a standin' offer of a present of a ten-dollar gold piece to any hired man who kills himself by workin' too hard."
From an untitled item in the [Great Bend, Kansas] Barton County Democrat (August 10, 1906):
Everet Strong has a hitch in his get along these days, having sprained his ankle in some manner Saturday.
From an unidentified item in the [St. Louis, Missouri] KATY Employes' Magazine (1913) [combined snippets]:
P. J. Cain, one of our dispatchers, was also in the Elks' Minstrel, but he says an Irishman never did make a good "nager." He might have done all right if he could have danced, but says he had a hitch in his get-along and was out of commission.
"KATY" was an acronym for the Missouri-Kansas-Texas Railroad.
From "A Redeeming Feature," in The Judge (New York, 1914) [combined snippets]:
"Dat mule o' mine," confessed old Brother Bulginback, "ain't much to look at, and, as a mattuh o' fact, sah, de varmint am dess as no 'count as he looks. He's a contaminated scoun'el, any way yo' 'proaches him. He's got a bad eye , a hitch in his git-along, and a disposition like one o' dese yuh white cuhnels wid de dyspepsy. he kicks—lammed me th'oo the side o' de shed dis mawnin'—and once in a while he nips a chunk out o' muh pussonality. ..."
From an advertisement for the Packer Produce Mercantile Agency, in the Chicago [Illinois] Packer (February 18, 1928):
A Hitch in the Git-Along
The sayings that denote trouble are many. | Down in Texas when a man i crippled they say, "He's got a limp in his trot." | In some other states they say, "He's got a hitch in his git-along."
...
Th firm which is suffering a hitch in its business "git-along" through doing business with some concern it knew nothing about knows what that expression means.
<>...
The time to back around and get out of trouble is before you get into trouble. Don't wait till you're "in tight." "got a hitch in your git-along," or a "limp in your trot. Order THE RED BOOK now!
From Helen Mowrey, "Miscellaneous Accounts," in the [Omaha, Nebraska] Union Pacific Magazine (March 1931):
Isabelle Harrington came back with a hitch in her get along after climbing "them thar" hills at Mt. Hood.
From A.J. McFadden, "California Agricultural Prorate Law—Difficulties of Enforcement," in Proceedings of the 67th Convention of California Fruit Growers and Farmers, December 18–20, 1934 (1935) [combined snippets]:
Possibly some of you, with a hard background of practical experience, have been troubled by a lurking suspicion that this state of affairs is too good to be true and that possibly all has not been told. I am not finding any fault with the speakers who have handled their subjects in this manner. It would be foolish foe any administrator not to make as good a showing as possible in the presence of the boss from back East and just as bad for the latter to admit the possibility that his brain child had a slight "hitch-up in his git along."
From an unidentified item in the [St. Louis, Missouri] Trans-communicator (1937) [combined snippets]:
Please note the "Bro." attached to E. J. Abbot above. I am very glad to greet him as such, and sorry I failed in last write-up (but yours truly had a hurt and spent a week in hospital, with a hitch in the git along), and also forgot that $9.00 was collected and remitted to the Postal Telegraph Company January 30th for birthday greeting to the President, and aid in his great fight against infantile paralysis.
From "The Hill Billies Go to Grand Lake," in the [Steamboat Springs, Colorado] Steamboat Pilot (July 27, 1939):
Well some of them [women in the group] were kind of jealous because Vera go to dance and we caught Cora making eyes and first thing you know a little wizened old guy comes up and says, "Lady, would you—" and that was as far as he got. He was going to ask for a match but Cora leaped up and grabbed him around the neck and dragged him out on the floor. The poor old guy had a hitch in his get along but that didn't bother Cora, she just limped a little and pretended that was the way they were dancing.
Conclusions
The expression "a hitch in [one's] git-along" began appearing in print in the United States by 1902. During the first half of the twentieth century, it seems to have had a particularly strong following among train men, telegraph operators, and others in the midwestern states, most notably Missouri, Kansas, and Colorado.