BURLINGTON, Vt.—Bernie Sanders, the independent Vermont senator who came up through that state’s democratic socialist party, launched his insurgent campaign for president four years ago as an outsider challenging the Democratic machine.

Now, in this early stage of the 2020 presidential race, Mr. Sanders is a Democratic front-runner and newly minted millionaire, with high name recognition and $18 million raised in the first six weeks of his campaign, so far outpacing his rivals. His campaign today has the trappings of the very...