In U.S. idiomatic usage, "the circuit" may refer generally to a fixed set of destinations on a performance tour. One early setting where the term was used is baseball, where "the circuit" refers to all of the cities or towns within a league where games are scheduled. Thus, from Athletic Sports in America, England and Australia (1889):
At the annual meeting in Cleveland, December 7th, the Athletic and Mutual Clubs were expelled from the league [the National League of Professional Baseball Clubs, formed earlier that year] because they failed to play out their scheduled games in the final series to be contested in the West. The Athletic and Mutual Clubs, realizing the weakness of the league circuit, and knowing that none of the clubs had made any money in 1876, did not believe extreme measures would be taken, but they failed to understand the determination of the men who had organized the league.
Baseball terminology has given us not only the phrases "the big-league circuit" (referring all Major League Baseball teams and their cities), "the senior circuit" (for the National League, the older of the two Major League Baseball leagues), and "the junior circuit" (for the American League, the younger "big" league), but also "the tank-town circuit" (referring to minor-league baseball leagues whose member teams play in towns so small that their most notable structure is their municipal water tank).
The phrase "the tank-town circuit" goes back to the the 1910s, first as a baseball term and later as a broader show-business term. The usage hasn't died out, as we see for example in this instance from Arne Lang, Prizefighting: An American History (2008):
Resuming his career, he [Jack Johnson] fought sporadically into the decade of the ¡930s, ending up as a trial horse [a name opponent for promising young boxers on the rise] on the tank town circuit. For every fight that he had in his professional dotage, at least that many were canceled, some at the behest of politicians yielding to pressure from the Ku Klux Klan.
Another "circuit" that authors sometimes cite today is the long-defunct vaudeville circuit. For example, from Tom Cutler, The Gentleman's Bedside Companion (2011):
Harry [Houdini] concentrated increasingly on escapes, which particularly suited his uncouth, bombastic performance style, and in 1899, at the age of 25, he was booked on the vaudeville circuit, where he became a huge hit.
The term "vaudeville circuit" refers to a rather vaguely defined series of theaters that performers would perform at during the heyday of vaudeville (1880s to 1930s). According to the Wikipedia article on Vaudeville, some entertainment entrepreneurs set up chains of theaters (called circuits) to simplify the task of booking acts that toured numerous cities:
By the late 1890s, vaudeville had large circuits, houses (small and large) in almost every sizable location, standardized booking, broad pools of skilled acts, and a loyal national following. One of the biggest circuits was Martin Beck's Orpheum Circuit. It incorporated in 1919 and brought together 45 vaudeville theaters in 36 cities throughout the US and Canada and a large interest in two vaudeville circuits. Another major circuit was that of Alexander Pantages. At his hey-day Pantages owned more than 30 vaudeville theaters and controlled, through management contracts, perhaps 60 more in both the US and Canada.
It seems probable that the establishment of theater circuits in the late nineteenth century led to popular use of the phrase "on the circuit" by performers in various entertainment professions—perhaps including sports—and by the journalists who wrote about them. Merriam-Webster's Eleventh Collegiate Dictionary (2003), which orders its entries by earliest occurrence, indicates that the sports sense precedes the song-and-dance sense:
circuit n ... 5 a : an association of similar groups : LEAGUE b : a number or series of public outlets (as theaters, radio shows, or arenas) offering the same kind of presentation
But both meanings seem to have arisen fairly close in time to one another.