There’s no formal mechanism for the Democratic Party to choose a new presidential candidate, but precedents and party rules guide what would take place.
In the event of a candidate dropping out of the race, or otherwise becoming incapacitated, that candidate’s nominating party decides on a new candidate. In the Democratic Party’s case, that would mean a vote on a new candidate by the four hundred forty seven members of the Democratic National Committee.
In the history of the United States, a major-party presidential candidate has never dropped out, and for good reason. Even if Clinton’s health issues lead to a precipitous slide in her support, the nomination of another candidate would be a PR disaster for the Democrats.
Barring a truly severe health episode, Clinton is very likely to remain at the top of the ticket.
There are also legal issues to contend with. Most states’ presidential registration deadlines have already passed, deadlines that would require a court order to overturn. Early voting has already started in some states, and it’s not clear how (or if) early ballots could be reassigned. Any attempt to replace Clinton would be fraught with thorny legal challenges.
(On the other hand, the leadership of the Democratic Party is apparently already hard at work working out the logistics of a Clinton dropout. Major Clinton backers have already started to tout that Clinton’s VP is ready to “take over” from her. A switch may be more likely than it would seem.)
And if there was a switch, Joe Biden would probably be Clinton’s replacement. There’s precedent for high-level elected officials to replace candidates after they drop out, and Biden is the highest-ranking elected official eligible for the presidency. Internal reports also have Biden pegged as the frontrunner, and he is very popular with the American public.
If Clinton isn’t replaced, her apparent frailty (compounded by her long history of health issues) is going to put increased emphasis on Tim Kaine.
We could see more attention paid to the Republican VP pick as well: Mike Pence, after all, was pitched as someone who could “fill in” for Trump’s weaknesses.
In an election between two obese elderly people, anything can happen. American voters are going to face big choices come November.
They just might not be the choices they expect.