Not illegal, sure, but the lack of warrant may (should) mean the information is inadmissible in court. It's no different than if they stole it.
It’s also relevant that ICE mostly deals in civil law, not criminal. While illegal entry is also a crime, deportation is a civil enforcement action. The fruit of the poisonous tree doctrine doesn’t quite apply. There might be something similar though, and I’d love someone more knowledgeable to chime in.
No, getting evidence voluntarily from any party (or even involuntarily and illegally from a third party!) doesn't trigger the exclusionary rule. No rights triggering the rule are violated in the voluntary case, and if the government violated A’s rights but then use the evidence only against B, the rule doesn't apply either.
And, of course, this is ICE, and the exclusionary rule only applies in criminal cases, not in deportation (civil removal) cases, so the question is largely moot, anyway, even if they were doing this for evidence rather than just to locate targets.
It's extremely different.
Law enforcement can ask to search in cases where they don't have a warrant or other legal reasons to insist on a search. If you say "yes", the results are admissible. People mess this up all the time when cops pull them over or knock on the door.
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.)