> If the information was the type covered by an expectation of privacy, then the lack of a warrant likely would make it inadmissible

In a criminal trial, of the person who had the legal expectation of privacy, which in most cases is the person with possession of the documents, etc., not the subject of the documents, where those are different.

But, in any cases, not in most civil proceedings, and explicitly not in civil removal (deportation) proceedings, which is what ICE is mostly concerned with here, so the Constitution is essentially irrelevant as there is no practical remedy available for violations (which is one reason states have adopted rules against information sharing with immigration officials.)