What should a sister do regarding her sister's request to contact a young man she used to speak with, knowing that the sister believes this communication is impermissible?

Do not reply to this young man's message, and cut off all means of communication with him or with any other non-mahram men. Beware of being lenient in such matters, as their consequences are grave. Advise your sister and remind her of Allah, and warn her of the repercussions of these forbidden relationships. If she does not desist, then inform someone who can prevent her, such as her father.