DES MOINES, Iowa—Wearing their team’s colors of blue and yellow, they arrived here from around the country at their own expense to support the Democratic presidential bid of Pete Buttigieg.

Barnstormers for Pete, a grass-roots effort organized online, helped give the mayor of South Bend, Ind., one of the biggest cheering sections at a closely watched Democratic Party event Friday. Members of the group are planning to show up again in Iowa and other states early on the nomination calendar.

“I...