The Republican National Committee sued the State Department Wednesday to try to get access to communications from former Secretary of State Hillary Clinton and her top aides, escalating the fight over Mrs. Clinton’s private correspondence as the presidential race kicks into high gear.

The RNC is suing over electronic communications from Mrs. Clinton and top aides when she was secretary of state, as well as communications between members of Mrs. Clinton’s presidential campaign and State officials after she left the department.

“Time is of the essence,” attorney Jason Torchinsky wrote in paperwork filed Wednesday with the U.S. District Court for the District of Columbia. “The information that the RNC seeks bears on Secretary Clinton’s fitness to serve as the next President of the United States and will have no value unless it makes its way into the public’s hands before the election.”

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The RNC is seeking communications from Cheryl Mills and Jacob Sullivan, who were top aides to Mrs. Clinton at the department, Patrick Kennedy, undersecretary for management at the State Department, and Bryan Pagliano, a former IT staffer who reportedly helped set up Mrs. Clinton’s private email server that ran out of her New York home.

Mr. Pagliano has reportedly been offered an immunity deal with the Justice Department in the ongoing probe into Mrs. Clinton’s email setup, though Attorney General Loretta E. Lynch would not confirm such a deal during testimony before Congress Wednesday.

The RNC also wants communications between State Department staffers and members of Mrs. Clinton’s 2016 presidential campaign, including campaign chairman John Podesta and communications director Jennifer Palmieri, since Feb. 1, 2013, when Mrs. Clinton left the State Department. They are also seeking communications since Feb. 1, 2013, between State Department staff and a domain tied to Mrs. Clinton’s private email account.

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The RNC said the department has been stonewalling their original Freedom of Information Act (FOIA) requests for the information filed last year, which they said prompted the lawsuits Wednesday.

“We’re looking for the State Department to comply,” RNC Chairman Reince Priebus said on Fox News. “She’s running for president. We have a right to the information, and we’re going to pursue it.”

State Department officials said Wednesday they don’t comment on pending litigation. Mrs. Clinton has said repeatedly she didn’t send or receive any material that was marked classified at the time.

But more than 2,000 of the approximately 30,000 e-mails released by the department under a court order have been retroactively deemed to include classified material.

“This frivolous lawsuit is just another attempt by Republicans and right-wing groups to try to hurt Hillary Clinton’s presidential campaign because she is the candidate they fear the most in a general election. It is not going to work,” Clinton campaign spokesman Brian Fallon said in a statement.

Asked about the lawsuit Wednesday, Mrs. Clinton’s top Democratic rival, Sen. Bernard Sanders of Vermont, said his concerns are elsewhere.

“There’s [an] investigation going on, there’s a process going on with emails … what I am focusing on is the issues impacting the American people,” Mr. Sanders said on MSNBC’s “Andrea Mitchell Reports.”

But a spokesman for the Democratic members of the House Select Committee on Benghazi said the lawsuit is further proof that congressional Republicans’ investigations into Mrs. Clinton’s emails and the Sept. 11, 2012, terrorist attack in Libya are political.

“It is far past time that Republicans stopped wasting taxpayer funds on partisan investigations that have served merely as an arm of the RNC, and let the RNC do its own dirty work,” the spokesman said.

News of Mrs. Clinton’s private email arrangement only broke in March 2015 after being uncovered by the committee’s investigation into the Benghazi attack.

A spokesman for the committee’s Republican members shot back that Democrats on the panel are only focused on politics and protecting Mrs. Clinton, the Democratic presidential front-runner.

“Their motivations are obvious: They’ve endorsed the former secretary of state for president, and will no doubt continue to act as a full-fledged arm of her presidential campaign,” the GOP spokesman said.