Florida Democrats have accused the swing state's Republican leadership of impinging on the fundamental rights of Americans amid growing voter anger at lengthy queues to vote, the shutting down of early voting and chaos in Miami over absentee ballots.

The state's Democratic party filed a lawsuit on Sunday to keep polling places open until election day as the Republicans stood accused of attempting to disenfranchise its opponents with new limits on early voting that contributed to waits of more than seven hours to cast ballots in Democratic strongholds such as Miami.

The Miami-Dade elections headquarters shut it doors on Sunday to people attempting to request absentee ballots because so many people showed up. Outside, would-be voters protested, shouting: "Let us vote".

Myrna Peralta, who waited with her four-year-old grandson for nearly two hours before being turned away, told the Miami Herald: "This is America, not a third-world country … They're not letting people vote."

After the outcry spread over social media, the department opened its doors again later in the afternoon. But the incident reflected deepening frustration at what are widely seen as Republican attempts to manipulate the election.

Long queues over the past week for early voting reflected a record turnout to vote early in Florida, with about 4 million people casting their ballots in advance of election day proper on Tuesday, as well as the longest ballot paper in the state's history, including complicated constitutional amendments.

But Democrats said they were also caused by Republican legislation sharply cutting the number of days for early voting from 14 to eight. That included scrapping voting on the final Sunday before election day, when large numbers of African Americans traditionally went to the polls after church.

The Florida Democratic party filed a legal action in Miami to try to keep at least some polling stations open for early voting in three key counties – Miami-Dade, Broward and Palm Beach – which account for about one-third of the party's support in the state.

The lawsuit said "inadequate polling facilities" contributed to long waits in all three places. The last voters in Miami cast their ballots at 1am on Sunday even though the polling place official closed at 7pm on Saturday.

"The extraordinarily long lines deterred or prevented voters from waiting to vote. Some voters left the polling sites upon learning of the expected wait, and others refused to line up altogether," the legal action said. "These long lines and extreme delays unduly and unjustifiably burdened the right to vote."

The state's Republican governor, Rick Scott, who has the power to extend early voting days, declined to do so.

Rod Smith, the chairman of the Florida Democratic party, accused Republicans of impinging on people's rights.

"Voting is a fundamental right, and we all have an interest in assuring that all Americans have effective opportunities to vote," he said.

"Florida's Republican state legislature has already reduced the number of days to early vote by six days. Because of Governor Scott's refusal to follow precedent and extend early voting hours in the face of unprecedented voter turnout in south Florida, we are requesting in federal court that more Floridians have a meaningful chance to early vote."

Separately, a judge ordered an extension to early voting at an Orange County polling station after it was shut by a bomb scare on Saturday.

The US justice department has one eye on the 2000 election debacle of hanging chads and discounted votes, as well as more recent Florida election legislation struck down in federal court as discriminating against minorities, so it is monitoring the conduct of the vote in Florida to determine if the long queues are a deliberate obstacle to voting and therefore in breach of the 1965 Voting Rights Act.