The Trump administration has put its search for the Justice Department’s No. 3 official on the back burner after failing to persuade several candidates to take the challenging position, according to people familiar with the matter, in the latest sign of the difficulties besetting the agency.

The department lacks permanent, politically appointed leaders to oversee at least five high-profile units, including the criminal, civil and tax divisions. And it has been a frequent target of attacks by President Donald Trump for what he claims are its shortcomings, including its handling of the probe into Russian interference in the 2016 election that he has repeatedly called a “witch hunt” and allegations of misconduct by the Federal Bureau of Investigation.

On Tuesday, Mr. Trump took aim again at his attorney general, saying on Twitter that someone in the post other than Jeff Sessions could have avoided “so much time and money wasted, so many lives ruined” by the Russia probe, which is being led by special counsel Robert Mueller and overseen by Deputy Attorney General Rod Rosenstein.

In the last week, Mr. Trump has tweeted about the Russia probe or the Justice Department more than 20 times. Mr. Sessions recused himself from the probe because he was a member of Mr. Trump’s 2016 campaign, which is being investigated for possible collusion with Russia. Mr. Trump has denied any collusion.

Without any immediate candidate ready to sign on to the job of associate attorney general, the administration is instead focusing on other vacancies. Department officials say the delays have strained resources and limited the agency’s ability to fully enact and implement new policies.