WASHINGTON — Senator Lisa Murkowski, Republican of Alaska, announced on Friday that she would seek re-election, formally entering what is expected to be the most expensive and challenging race of her political career after voting to impeach former President Donald J. Trump.

Of the seven Republicans who found Mr. Trump guilty of incitement of an insurrection after the Jan. 6 Capitol riot, Ms. Murkowski is the only one facing re-election in the 2022 midterms. A moderate who has never won a majority of the general election vote in her Republican-leaning state, she is seen as the G.O.P.’s most vulnerable Senate incumbent at a time when there is little tolerance among the party’s core supporters for criticism of the former president or cooperation with President Biden.

The race sets up a proxy battle between Mr. Trump, who has endorsed a Republican challenger, and Senator Mitch McConnell of Kentucky and other Republican leaders, who are backing Ms. Murkowski in her bid for a fourth full term.

In a campaign video announcing her re-election bid, Ms. Murkowski made no mention of Mr. Trump, instead highlighting her work on behalf of the state and offering a pointed warning that “lower-48 outsiders are going to try to grab Alaska’s Senate seat for their partisan agendas.”

“I’m running for re-election to continue the important work of growing our economy, strengthening our Alaska-based military and protecting our people and the natural beauty of our state,” Ms. Murkowski, a third-generation Alaskan, said. “I will work with anyone from either party to advance Alaska’s priorities.”

Ms. Murkowski, first appointed to the Senate in 2002 by her father after he became governor and resigned from the seat, is the second-most-senior Republican woman, after Senator Susan Collins of Maine, who won her own costly re-election bid last year. Ms. Murkowski has established herself as a crucial swing vote with strong relationships in both political parties, most recently helping negotiate the $1 trillion bipartisan infrastructure bill that Mr. Biden is expected to sign into law next week.

It is unclear, however, whether Ms. Murkowski’s record of directing aid and support to her state will be enough to overcome the grip of the former president on her party. Alaska’s Republican Party censured her in March for voting to convict Mr. Trump. The former president endorsed a primary challenger, Kelly Tshibaka, the former commissioner of the Alaska Department of Administration, who has promoted false theories of voter fraud in the 2020 election and hired a number of former Trump campaign staffers.

“Lisa Murkowski is bad for Alaska,” Mr. Trump said in a June statement, after Ms. Murkowski voted to confirm Deb Haaland as Interior secretary. “Murkowski has got to go!”

The National Republican Senatorial Committee, the campaign arm for Senate Republicans, has backed Ms. Murkowski, who also has support of her party leadership. Ms. Murkowski has a long record of bucking her party, having helped to shut down the effort to repeal the Affordable Care Act, opposed the confirmation of Justice Brett Kavanaugh to the Supreme Court and voted for a number of Mr. Biden’s nominees during the first year of his administration.

“We support all of our incumbents, and fortunately for us, we’ve got great candidates running in our primaries,” said Senator Rick Scott of Florida, the chairman of the organization, on NBC’s “Meet the Press” on Sunday. But in the same interview, he acknowledged “you’d be foolish not to want and accept Donald Trump’s endorsement.”

In 2010, Ms. Murkowski lost a primary race to Joe Miller, a Tea Party candidate, but mounted a successful write-in campaign, becoming the first write-in candidate in more than 50 years to win an election. In January, she told reporters that she would not switch parties, even as she questioned whether she belonged in a Republican Party that was influenced so heavily by Mr. Trump.

“As kind of disjointed as things may be on the Republican side, there is no way you could talk to me into going over to the other side,” Ms. Murkowski said at the time. “That’s not who I am; that’s not who I will ever be.”

Under a new election system approved a year ago, Ms. Murkowski will first compete in an open primary where the top four candidates will then advance to a ranked-choice general election.