The Netroots Nation convention in Philadelphia showcased the far-left energy of the Democratic Party, but that enthusiasm was not for the party’s presidential front-runner, Joseph R. Biden, with some diehard activists threatening to vote for a third party or stay on the sidelines in 2020 if he captures the nomination.

The threat highlights a rift in the party between the establishment embodied by Mr. Biden and House Speaker Nancy Pelosi and a new brand of Democrats who demand a bold, socialist-style agenda.

“The only Democrat I’m prepared to vote for is Elizabeth Warren. I’ll vote third party before I vote for Biden,” said Michigan voter Brent Simmons, 70, a retired civil rights lawyer with the NAACP.

“We’ve got to break with New Democrats/Clintonian Democratic Party. Nancy Pelosi is a part of that. It’s time for a new generation to move into leadership positions,” he said.

Ms. Warren, a Democratic senator from Massachusetts, was the hands-down favorite of liberal activists at the Netroots Nation convention last weekend.

The crowd gave Ms. Warren a thunderous standing ovation, the loudest at the often raucous event when she promised a Justice Department probe of President Trump’s treatment of immigrants who are in the U.S. illegally.

“You abuse immigrants, you physically abuse immigrants, you sexually abuse immigrants, you fail to get the medical care that they need, you break a law of the United States of America. Donald Trump may be willing to look the other way, but President Elizabeth Warren will not,” said Ms. Warren, setting off the cheers.

She has been climbing in national polls and vying with Sen. Bernard Sanders of Vermont and Sen. Kamala D. Harris of California for the No. 2 position in the race.

“I love Joe Biden but I understand the torch does need to be passed,” said Kevin Harmer, 28, a liberal political consultant in Philadelphia. “We might need a new party. I’m not a fan of the two-party system.”

He said he would hold his nose and vote for Mr. Biden as the Democratic nominee. But he couldn’t guarantee the rest of the crowd at Netroots Nation would do the same.

Mr. Biden did not attend the convention. During his tenure as vice president, he had clashes with the activists at Netroots Nation and didn’t need a repeat during the campaign.

He also has made little effort to reach out to his party’s far-left wing. Instead, he has kept the campaign focused on moderate Democrats and winning back rural and Rust Belt voters who backed Mr. Trump last time.

In 2016, the same split between left-wing and moderate factions contributed to Democratic nominee Hillary Clinton’s stunning loss. The divide was exasperated by the Democratic National Committee rigging the primary process against Mr. Sanders, which infuriated his supporters and soured them on Mrs. Clinton.

Democratic Party officials have labored this campaign season to avoid the appearance of picking a favorite in the crowded field. Party officials also say they are confident that the fierce opposition to Mr. Trump will be enough to unite the party’s voters behind its nominee.

Still, the concerns about an insurgency against Mr. Biden or another mainstream candidate remains a real issue.

“It definitely concerns me,” said Rachel Earnest, a Democratic Party operative in Pennsylvania. “There is a certain element of disestablishmentarianism on the left. Some of us have difficulty in our regular life just going with the flow.”

She said the party’s liberal activists, which she counts herself among, need to get the message that demanding ideological purity could have dire consequences.

The midterm elections in 2018 “should have taught the lesson that center-left candidates get elected. They do get elected, as much as I disagree with their politics,” Ms. Earnest said.

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