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“The app that was intended to relay caucus results to the party failed; the party’s backup telephonic reporting system likewise has failed,” wrote Mr. Biden’s general counsel, Dana Remus. “Now, we understand that caucus chairs are attempting to — and, in many cases, failing to — report results telephonically to the party.”

The campaign asked to see the “methods of quality control” being used by the party and requested “an opportunity to respond, before any official results are released.”

Even before Monday, there were other concerns with the app itself, which was developed by a private firm called Shadow. Cybersecurity experts worried that it had not been vetted, tested at scale, or even shown to independent experts before being introduced in Iowa.

Christopher C. Krebs, the director of the Homeland Security Department’s cybersecurity agency, said late Monday that the mobile app had not been vetted or evaluated by the agency.

The candidates decided not to wait for any results, one by one giving variations of a victory speech, beginning with Senator Amy Klobuchar, though none were quite as bold in their proclamations of success as Pete Buttigieg, the former mayor of South Bend, Ind.

“Tonight, an improbable hope became an undeniable reality,” Mr. Buttigieg declared, as a campaign spokesman spent the evening posting screen shots of precincts where the candidate had won.