First, find out where the person is
Two official search systems exist in the United States, they are free, and they are the ones lawyers use. Neither requires a lawyer to search.
A person who has already been flown out of the United States will not appear in either system, because both search custody inside it. That absence is information: it usually means the person has been removed, and the question becomes which country received them.
Getting a message to someone
The Red Cross and Red Crescent movement runs a tracing network for exactly this situation. It is called Restoring Family Links, it operates in almost every country, and it exists to reconnect people separated by detention, conflict and migration and to carry messages between them. It is free, and a family can start a trace from their own country rather than from the country the person was sent to.
What a family can do from outside
Three things are worth understanding early, because they change what is possible later.
The first is documents. If the person held a grant of withholding of removal or of protection under the Convention against Torture, that grant is a finding by a United States immigration judge and it sits in the immigration court record. A family that can find the file number, the court, and the date is holding the most useful thing anyone can hold.
The second is that a record with a date on it is the beginning of everything. A complaint made in writing to a national human rights commission, a request to a consulate, a note of the day and time a call was refused: these are worth more later than anyone expects at the time.
The third is that the country matters enormously. Whether an office of the United Nations refugee agency exists there, whether the country accepts individual complaints, whether the person’s own embassy is even in that country, all differ. The country guides set this out one state at a time.