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I remember reading that the US president can give an order to a civilian via some special letter. (The example given involved an IT company employee ordered to install a back-door in the company's infrastructure.) However, I cannot find this described in the Wikipedia article Powers of the president of the United States. Is my memory failing me, or is this case missing from the Wikipedia article?

Generally, in times of peace the President can not give orders to a private citizen.

However in times of war or emergencies, the courts have allowed the President to give some pretty extreme executive orders as outlined in the article you mentioned. https://en.wikipedia.org/wiki/Powers_of_the_president_of_the_United_States#Emergency_powers

Short of those extraordinary measures, there is not really a way for the President to personally order a civilian.

The executive branch of the US government, or specific parts of it, can demand that a person do particular things, when a statue has authorized such a demand. Such demands are not usually made at the level of the President of the United States, but the president could order a specific official to take such action. For example a National Security Letter orders a person to turn over specific information, and not tell anyone about doing so, as described in this article from The New Yorker.

There are other circumstances in which an executive branch government official may order a private citizen to take specific action. Indeed, this happens in almost every case of a police arrest, except those done by an arrest warrant (which is a court order).

During the Korean War, President Truman issued an order seizing a number of steel mills, to be operated by the government, in order to stop a strike which was, he said, impeding the military effort and thus the security of the United States. This order was challenged in the case of Youngstown Sheet & Tube Co. v. Sawyer, 343 U.S. 579 (1952) and was overturned on the ground that the President was not only not-authorized by any law, but had failed to follow the provisions of a law dealing with situations of the kind at issue. In that case the Court made it clear that such a seizure would have been permitted if done in conformance with a statute, and might have been permitted had no law dealt with such cases.