Is what the suitor did, proposing to an orphan girl who was raised by her sister and her sister's husband, with his knowledge that she is not their biological sister, considered Haram? And does it entail a specific action?

There is no objection to marrying the aforementioned girl as an act of kindness towards her, but marriage is not valid without a wali (guardian). The order of guardians is: the father, then the paternal grandfather, then the son, then the full brother, then the half-brother (from the father's side), then their sons, then the paternal uncles. If there is no qualified wali to arrange the marriage, then a Muslim judge may marry her off with her permission. If a judge is unavailable, then a just man may marry her off with her permission.

Know that whoever attributes a girl to himself as his child has committed a forbidden act. This adoption does not create mahramiyyah (prohibition of marriage) or inheritance. It is obligatory to nullify this lineage and inform the girl of her true lineage. It should also be noted that interaction between non-mahram men and women – even if for the purpose of marriage – is not approved by Islamic law.