While not a specific answer to the question, for context, it is worth noting that election fraud was widespread in the late 18th century, the 19th century and the early 20th century in the U.S., and was often successful. The electoral process (and the legislative process as well) was sloppy, chaotic, unruly and disorganized in that era and easily manipulated as a result.
In urban areas (Chicago was particularly notorious), immigrant dominated "political machines" were often involved.
In the South, violent voter suppression and official misconduct were often at fault. As explained, for example, here:
Black male suffrage became national in 1870 when the Fifteenth
Amendment prohibited states from discriminating against potential
voters because of race or previous condition of servitude (but not
sex). . . . For most of Reconstruction, blacks voted and often used
their resulting political power to protect their other rights.
Although white Democrats overthrew Reconstruction by forcibly keeping
blacks from the polls, thereafter they generally eschewed violence.
Instead, they relied upon discriminatory apportionment and election
laws to limit black and poor white political influence, and in the
1890s and 1900s, imposed poll taxes and literacy and property
requirements through constitutional revisions. Thus they disfranchised
virtually all black men and many poor whites, and thereby ensured
Democratic hegemony.
In that era, Congress frequently exercised its power to resolve disputed election for House and Senate and usually did so on a purely partisan basis, often allowing the wrong person to take office. From 1789 to 1908 the House of Representatives resolved 382 contested elections (keep in mind that the House of Representatives started out with just 65 members), of which only three were given to the minority party candidate in the House. The first such contest was in 1789 (in that case, upholding the election result), in the first first Congressional election under the current constitution. Since 1910, House of Representatives resolution of contested elections has grown much more rare, and state elections officials and courts almost always resolve the disputes before the House acts.
A more in depth review of the history of those disputes can be found in a 2007 academic journal article which looks at the question of why contests were initiated in the first place, rather than simply how they were resolved. As its abstract explains:
Recent studies of contested elections in the House have pointed to
party electoral goals as motivating their resolution in Congress.
However, little systematic research has been conducted on why such
elections were contested to begin with. Using historical data and new
statistical analysis of such elections from the late nineteenth
century, I find that, in contrast to the claims of some scholars,
political principles as well as electoral objectives mattered to
parties seeking to contest elections. In addition, election
conditions, particularly the means by which southern white Democrats
attempted to repress the vote of southern blacks, independently
influenced the probability of contestation. This finding has
implications for our understanding of Republican Party strategies and
electoral conditions in the South during the period, and the origins
of contested elections at other times in American history.
In this era, Congressional elections were somewhat ad hoc, as were Presidential elections in the late 1700s and early 1800s. While the U.S. Constitution authorized it to do so in 1789, Congress only passed a law providing for a uniform nationwide date for choosing Presidential electors in 1845. This law did not affect election dates for Congress, however, which remained within the jurisdiction of State governments, but over time, although the States moved their Congressional elections to this date as well. As late as 1876–77, there were still 8 states with earlier election dates, and 1 state with a later election date.
There were no U.S. Senate elections, of course, until after the 17th Amendment was adopted in 1913, so few of its elections occurred when Congressional resolution of disputed elections was common. There have been 25 election contests in the U.S. Senate since its members began to be elected, 3 of which were successful and 22 of which determined that an incumbent whose re-election was supported by state officials retained his seat. (There were seven cases during the era of legislative appointment, however, where a legislative appointment was rejected as invalid or the state's purported nominee resigned in the face of a challenge.)
The electoral process that resulted in the adoption of the Texas Declaration of Independence in 1836 and formed the Republic of Texas was illegal under the then applicable law of Mexico, and was basically fraudulent.
The election of about 40 men in 1836 that gave rise to the very short lived California Republic was likewise basically fraudulent, although elections were secondary in the U.S. annexation of California which was part and parcel of a large war between the U.S. and Mexico that started in connection with Texas, that was underway before and after California's annexation.
The 1854 election in Kansas over whether it would be a free state or a slave state mentioned in the answer of @Ryan_L is another example from the same era.
The outcome in the Hayes-Tilden Presidential election of 1876 was probably a case of election fraud (see also here).
The sequence of events by which the Kingdom of Hawaii adopted its Constitution of 1887 through its annexation to the U.S. in 1898 involved multiple instances of what amounted to election fraud and other unlawful conduct by white businessman who had relocated to the islands. The U.S. government formally apologized for this in 1993.
The dynamics of the election process were also influenced in the early U.S. by the fact that ballots were not secret, although that goes more to duress than outright fraud and may have actually reduced election fraud. In the United States, most states had moved to secret ballots soon after the presidential election of 1884. However, Kentucky was the last state to do so in 1891, when it quit using an oral ballot. Therefore, the first President of the United States elected completely by secret ballots was president Grover Cleveland in 1892.
Needless to say, oral voting was also not auditable in the way that paper ballots are auditable.
In part, this happened because the federal government wasn't taken as seriously and wasn't as important in the early United States. According to political scientist Randall B. Riley, in his book "Congress: Process and Policy":
Until the 1880s the length of service of the average senator and
representative remained at a low and fairly constant level --
representatives averaged two years (after a higher level of around
three years during a peak of House influence on national policy in the
1810s and 1820s) and Senators around four years. Members of both
chambers, especially the House, routinely left Congress for other
opportunities, both governmental and private. Mid-term resignations
were common.
Beginning about 1880 the average years of service rose dramatically in
both houses, doubling in the Senate [to eight years] in less than two
decades and almost tripling in the House [to six years] in three
decades. The upward trend continued, with breaks for political
turnovers, until the late 1960s.
Sessions of Congress were in any case short in the early Congresses because they had little to do. The first Congress met seventeen months out of twenty-four implementing the new constitution. The Civil War and Reconstruction Congresses met between ten and twenty-two months. In every other two year session of Congress until the one commencing in 1911, Congress met for less than twelve of the twenty-four months of the session.
And, in the 19th century, Congress was a tough place. Physical violence on the House floor was not infrequent and guns and knives were occasional carried onto the floor. As an October 22, 2018 article in The Atlantic magazine recounts:
In the three decades before the Civil War, members of the House and
Senate routinely threatened each other with violence, and often acted
on it too. They brawled on the House floor; they faced off in duels;
they fired shots in Congress. They beat each another senseless with
canes. All told, members of Congress engaged in at least 80 acts of
physical violence between 1830 and 1860, a remarkable fact uncovered
by Joanne B. Freeman, a historian at Yale, in her superb new book The
Field of Blood.
