Jessica Flannery spent the first half of March in Democrat Hillary Clinton’s Des Moines campaign office with three dozen other volunteers, calling the front-runner’s supporters in Iowa to ensure they showed up at county conventions and were elected delegates to the national party gathering in July.

Republican front-runner Donald Trump’s campaign, meanwhile, sent an email to local Republicans just before the statewide county convention inviting them to sign up to be a delegate for him. The campaign followed up a few days later...