WASHINGTON—Vermont Sen. Bernie Sanders joined the burgeoning field of candidates seeking the Democratic presidential nomination, vowing Tuesday to take on “the powerful special interests that dominate our economic and political life.”

Mr. Sanders, who battled Hillary Clinton for the nomination three years ago and helped reshape the party, will return to the presidential stage with a loyal group of supporters and arguably the strongest online fundraising machine in Democratic politics. But unlike 2016, he will need to compete...