On-the-ground organization is hardly a guarantee of gaining the nomination: In 2016, Donald Trump rejected traditional organizing, while one of his most persistent foes, Ted Cruz, had an extensive and innovative operation in Iowa. Mr. Cruz won the state’s Republican caucuses, but was soon overtaken by Mr. Trump.

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But early organizing can be critical for get-out-the-vote operations and is important for candidates who want to capitalize on any bursts of momentum.

In Iowa and New Hampshire, early impressions of the two dozen candidates’ ground-game operations are beginning to form. Here are five takeaways about the organizational state of play for the leading candidates, based on conversations with party activists, campaigns and voters:

Biden has work to do

The former vice president enjoys decades-long friendships with party leaders in both states, and he has accelerated his campaign travel, making an extensive swing through Iowa and New Hampshire in the last week. Multiple Iowa trips are planned for August.

But Mr. Biden starts from behind organizationally. He entered the race at the end of April and began with a lighter public schedule than many of his opponents, allowing other earlier-launching campaigns to lock down experienced talent and build more visible volunteer operations first.

In Iowa especially, the impatience with his efforts among some activists was palpable this month following Mr. Biden’s shaky debut in the first presidential debate.