According to the controversial unitary executive theory, the President has sole authority over the Executive branch. Since the Department of Justice is within this branch, he could then direct the Attorney General to initiate investigations. Many of Trump's actions during his administrations, especially those during the first few months of his second term, are based on this theory. However, this is a markedly controversial theory, and many conclude these powers, their scope, and their strength, are not so absolute.
According to the New York Times:
Mr. Trump is not barred legally from directing the Justice Department to open investigations. But after the Watergate scandal, the United States developed a constitutional norm of law enforcement officials making investigative decisions independently of the White House.
What it's referring to regarding the Watergate scandal is the Saturday Night Massacre, when President Nixon ordered the DOJ to fire special prosecutor Archibald Cox, who was investigating the President's role in the Watergate affair. The AG and other top DOJ officials resigned rather than carry out this order.
Since then, there has been a tradition of independence between the White House and the DOJ, but there's nothing requiring it.
It doesn't matter whether Trump's complaint has legal merit. Under this theory, he could direct the Justice Department to investigate, and they should independently decide whether there have been chargeable offenses (and whether it's worth trying to bring an indictment).
Note that in this case, I don't think he's ordering an investigation into Springsteen himself. He's ordering an investigation into the Harris campaign for paying Springsteen for an endorsement. He's saying that this is a violation of federal election laws -- celebrities can endorse candidates voluntarily, but they can't be paid for endorsements. He's also claimed in the past that Oprah Winfrey was paid by the Harris campaign.
HOW MUCH DID KAMALA HARRIS PAY BRUCE SPRINGSTEEN FOR HIS POOR PERFORMANCE DURING HER CAMPAIGN FOR PRESIDENT? WHY DID HE ACCEPT THAT MONEY IF HE IS SUCH A FAN OF HERS? ISN’T THAT A MAJOR AND ILLEGAL CAMPAIGN CONTRIBUTION? WHAT ABOUT BEYONCÉ? …AND HOW MUCH WENT TO OPRAH, AND BONO??? I am going to call for a major investigation into this matter. Candidates aren’t allowed to pay for ENDORSEMENTS, which is what Kamala did, under the guise of paying for entertainment.
If targeting celebrities who endorsed political opponents had been attempted by another President, Trump probably would have accused them of "weaponizing" the Justice Department.