Who would investigate? The FBI? The Capitol Police? The Keystone Kops?
There is a Supreme Court Police force.
Also the President could issue an EO for the Secret Service to protect the Court and investigate any crimes committed there.
“The Supreme Court of the United States Police is a federal law enforcement agency that derives its authority from 40 U.S.C. ยง 6121. The Supreme Court Police enforce federal and District of Columbia laws and regulations, as well as enforce regulations governing the Supreme Court Building and grounds prescribed by the marshal and approved by the chief justice of the United States. The department’s mission is to ensure the integrity of the constitutional mission of the Supreme Court of the United States by protecting the Supreme Court, the justices, employees, guests, and visitors.” Wikipedia
I think the only one with authority is John Roberts. I wouldn’t want any of the alphabet agencies or corrupt politicians to investigate SCOTUS.