Glitches to a mobile app and a backup phone-in plan for reporting results in Iowa’s Democratic presidential caucuses delayed the outcome and heightened concerns about the integrity of the voting process in the 2020 elections.

After Monday night’s failure, some Democratic Party officials in the 1,700-plus caucus sites complained about the rollout and difficulties in downloading and using the app. Some said a final version was only made available this past weekend. Still, others said that when they tried to phone results to...