Report it to the authorities first, and early. There is no waiting period for a missing person in an emergency, and the official channels are the ones connected to hospitals, shelters and registration lists.
Write down what you know while you remember it. Full name, date of birth, what they were wearing, where and when they were last seen, any medical conditions or medicines, and a recent photograph. Under stress these details blur within hours.
Check the official lists, not only social media. Hospitals, reception centres and shelters register arrivals, and national authorities and recognised humanitarian organisations run family tracing.
Be careful with public appeals. They are sometimes the right thing and they carry real costs: a public post reveals that somebody is alone or vulnerable, attracts people who exploit that, and cannot be withdrawn once it has spread. Consider what it exposes, and prefer official channels where they exist.
Keep one place for information. Notes of who you contacted, when, and what they said. In a search that lasts more than a day, that record is what stops the same ground being covered five times.