1921-Vice President Roosevelt contracts polio while vacationing and resigns the Vice Presidency to focus on his recovery.
Let's say that he doesn't resign, but by 1924 his handicap forces his withdrawal from the ticket.
1924-President Wilson decides to run for reelection.
For a third term? I don't think so.
He is confident that he will have the support of the west due to the farm support and public power bills he's enacted, so he decides to go with an eastern for his veep and selects Secretary of Commerce Kennedy. (The booming economy has boosted Kennedy's profile over the last 3 years.)
Let me add a tweak here: Kennedy, a Catholic, could alienate the South... But Wilson, a Southerner and flaming racist, as the Klan et al knew quite well, is at the head of the ticket, so the effect is minimal.
Or a different tweak, assuming Wilson retires as seems likely to me:
The Democrats nominate Wilson's son-in-law William Gibbs McAdoo, despite his Klan affiliations; they put Kennedy in as VP to balance the ticket and reassure northern Catholic voters.
Wilson's health does not last out the campaign, and in late October 1924 he suddenly dies.
Another tweak: Wilson's health barely lasts out the campaign, and in November 1924 he suddenly dies - after winning the election. After some pulling and hauling, the Democrat electors vote for Kennedy for President.
Or with the second tweak, something happens to McAdoo after the voting.
Although one could say that in such a case, Kennedy is not actually elected, but
de facto succeeds to the Presidency.
Direct election of Kennedy in 1924 looks like ASB territory, more or less. He is Catholic, only 36 years old, and has no political experience; nor, AFAIK, any ambitions for political office.
However, I'll make one stab at it. I'll assume that Kennedy was personally opposed to Prohibition, though he probably avoided any public statements.
Suppose however that he was active in opposition to Prohibition in 1918-1919. It goes through anyway, with the result of the bootlegger boom and massive gangsterism. A Repeal League gets formed, supported by businessmen and reformers repulsed by gangster profiteering and violence, and the hypocrisy of "prominent" men who endorse Prohibition while patronizing bootleggers.
The reapportionment of 1920 gets done, unlike OTL, shifting power toward urban areas.
Harding survives his 1923 illness. However, he is dogged by the corruption of his administration.
As in OTL, the Ku Klux Klan grows to massive national importance (though it was not especially significant in much of the South at this time, and much of its numbers elsewhere came from populist discontent).
The 1924 election is chaotic. The Democrats nominate William McAdoo, endorsed by the Klan. The Republicans renominate Harding, with reluctance. There is an insurgent ticket of "Progressives" (as in OTL); and finally, the Repeal League mounts a campaign in combination with anti-Klan Democrats, and pick Kennedy for some reason. (Though I'm not sure what good electing a 'wet' President would do toward achieving Repeal.)
Harding and the Republicans are damaged when a scandal sheet publishes an interview with the White House bootlegger. This lends the Repeal Democrats some credibility. It also pushes dry Republicans to the Progressives, who remain adamantly Prohibitionist.
In the four-way scrap, Kennedy finishes second in electoral votes as the election goes to the House. Southern Democrats, Repeal Democrats, and
'wet' Republicans in the House elect Kennedy.