Gov. Mike Dunleavy has accepted Donald Trump’s endorsement for his 2022 reelection campaign, telling the former president he will not support Lisa Murkowski in her reelection bid for the U.S. Senate — a condition of winning Trump’s endorsement.

The former president has vowed revenge against Murkowski and other Republican lawmakers who supported impeachment for Trump’s role in instigating last January’s insurrection at the Capitol during certification of Joe Biden’s election as president.

Trump has endorsed Murkowski’s primary challenger, Kelly Tshibaka, a top appointee in the...