Privateering was declared illegal by the Declaration of Paris of 1856, but the United States government didn't sign it. The Confederate States of America (obviously) didn't either. The Confederacy did operate a modest privateering campaign against the Union during the Civil War, but it didn't contribute significantly to the outcome. I'm not aware of any similar activities by the Union.
Logically, the U.S. government probably refrained from issuing letters of marque during the Civil War because, technically, it wasn't a war. The Constitution provides that the United States enters a state of war when Congress passes a declaration of war. There was no declaration of war against the Confederacy, because the United States never recognized the Confederate States of America as a nation. (Neither did any other country.) In the official eyes of the U.S. government, what happened between 1861 and 1865 wasn't a war; it was the suppression of a rebellion. A letter of marque was an official document authorizing a private citizen to capture merchant vessels flying the flag of a nation with which the issuing government was at war. Since the U.S. wasn't at war with an enemy nation between 1861 and 1865, the Union government couldn't issue letters of marque. That, at any rate, is my reading of the situation.
Lots of people don't seem to realize it, but officially the United States has only gone to war five times: the War of 1812, the Mexican War, the Spanish-American War, World War I, and World War II. The various other armed conflicts in which American military forces have participated were not, according to the Constitution, wars.
I'm not sure when (or if) the United States officially took the trouble to make privateering an illegal activity for its citizens. But privateering certainly had disappeared by the time the U.S. next got into a war with a foreign power, in 1898.
Schoonerbum's discussion of the Lindberg "Independence War Schooner" is dead on target. It's a reissue of an extremely old Pyro kit, which was in turn pirated from the wood Model Shipways revenue cutter Roger B. Taney. It still makes a fine project for beginners - though it certainly shows its age. Retrofitting it as a War of 1812 vessel would take a bit of work, though.
Many, many years ago, Aurora made a kit called "Privateer Corsair." The instructions were extremely vague in their explanation of what it was, but it looked a good deal like the famous War of 1812 American privateer Prince de Neufchatel. It was a pretty little model (about a foot long, if I remember right), with relatively few pieces but reasonably nice detail for its age (the mid-fifties) - certainly a major improvement over its Aurora predecessors, the "Pirate Ship" and "Viking Ship." I've seen it on e-bay a couple of times. That kit would be a good basis for a model of a War of 1812 privateer.