WASHINGTON—Pete Buttigieg, mayor of South Bend, Ind., population 102,000, had no national network of wealthy political donors when he began his long-shot presidential run in January.

Not six months later, he is one of the strongest fundraisers among the 24 Democrats seeking the nomination, saying in an email Monday to supporters that he had raised $24.8 million over the past three months, adding to his $7 million total from the first quarter.

Mr....