Conway did not show up at a hearing Wednesday, after the White House said Monday it would not allow her to appear . The Democratic-led panel voted 25-16 to issue a subpoena.
Rep. Elijah Cummings, D-Md., the panel's chairman, said that Conway's actions were a clear-cut violation of the law and that President Donald Trump should fire her.
"This is about right and wrong. This is about the core principle of our democracy that no one in this country is above the law, not even Kellyanne Conway," Cummings said.
The U.S. Office of Special Counsel, which is unrelated to special counsel Robert Mueller's office, said earlier this month that Conway has been a "repeat offender" of the federal Hatch Act by disparaging Democratic presidential candidates while speaking in her official capacity during television interviews and on social media.
Federal law prohibits executive branch employees from using their official authority or influence to affect the result of an election. Trump and Vice President Mike Pence are exempt from the Hatch Act, but there are no exceptions for White House employees.
Republicans said Democrats were trying to curb Conway's free speech.
Conway, a frequent presence on radio and TV, is being "targeted because she is good at what she does," said Ohio Rep. Jim Jordan, the panel's senior Republican.
Instead of focusing on issues that matter to Americans, Democrats "want to focus on Kellyanne's tweets," Jordan said, noting that senior advisers to former President Barack Obama also frequently appeared on TV and offered opinions on political matters.
"What we have here is a political spectacle," said Rep. Mark Meadows, R-N.C.
Cummings responded that Republicans had "gotten to the point, sadly, where breaking the law is OK."
"It is not breaking the law," Meadows shot back.
Special Counsel Henry Kerner, a Trump appointee, said the Hatch Act's restrictions on political activity were reasonable. Without the law, partisan actions by government employees "could quickly multiply and severely damage the public's confidence in the nonpartisan operation of government," he said.
Referring to Conway, Kerner said her "egregious and repeated Hatch Act violations, combined with her unrepentant attitude, are unacceptable from any federal employee, let alone one in such a prominent position."
Conway's conduct "hurts both federal employees, who may believe that senior officials can act with complete disregard for the Hatch Act, and the American people, who may question the nonpartisan operation of their government," Kerner said.
The agency does not have the authority to fire Conway, so it would be up to the president to follow Kerner's June 13 recommendation that she be dismissed. The recommendation marks the first time the watchdog office has recommended removal of a White House official over Hatch Act violations.
Trump has said he will not fire Conway , one of his most unwavering defenders. Conway is known for her fiery television appearances in support of Trump and his policies. She helped him win the presidency in 2016 as his campaign manager.
Conway has dismissed reports of possible Hatch Act violations, saying critics were merely trying to silence her because "they don't know how to beat (Trump) at the ballot box."
Instead of testifying to Congress on Wednesday, Conway accompanied Trump to a Washington hotel, where he was addressing the Faith & Freedom Coalition's annual policy conference.
In a related development Wednesday, House and Senate Democrats introduced a bill granting the Office of Special Counsel authority to bring complaints against high-level presidential appointees to the federal Merit Systems Protection Board, a quasi-judicial agency that oversees executive branch employees. The bill would allow the merit board to impose fines of up to $10,000 for each violation.
Conway's "flagrant violations of the Hatch Act and the president's refusal to terminate her highlight a glaring failure in our system of oversight of federal employees," said Rep. Jackie Speier, D-Calif. "If Ms. Conway wants to provide political spin on TV, she has the right to do so - but not on the taxpayer's dime."
The proposed legislation "gives federal ethics law teeth by imposing fines against those found in violation of the Hatch Act," she said.
Associated Press writer Darlene Superville contributed to this report.
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