Yes indeed the US has commissioned Privateers by Letters of Marque on two main occasions.

The first was the Quasi War with France in the Caribbean in 1798-1800. However the relatively restricted in time, space, and scale of that conflict meant that only a few score really got into the action.

The far larger case was the War of 1812 when several hundred vessels of all sizes were commissioned. They sailed from every port from Maine to Georgia, and were instrumental in capturing the majority of the 1,500 or so British vessels taken during the war. And though by 1814 the British bloackade of the American Coast had become sufficiently strong to greatly reduce their numbers and freedom of movement, Privateers never stopped putting to sea completley and still made their mark. By taking prizes they could also impact the British war effort, forcing the diversion of ships for convoy duty and escort, driving up insurance rates in Britain, and diverting other resources. For instance one of the main reasons that Baltimore became the second target of the joint Army-Navy force under Ross, Cochrane, and Cockburn after they burned Washington, was the large number of Privateers based out of Baltimore.

In the end somewhere upwards of 500 vessels were used as Privateers, each with a commission from President Madison after Congress delegated to him the authority to issue them early in the conflict. Most were very much on the smaller side though, between 100 and 300 tons displacement and no more than 12-14 guns with lighter armed and smaller ships aplenty. Meanwhile regular naval ships were generally much bigger, the modest sloop USS Wasp was 450 tons for instance and the big 44 gun super frigates Constitution, President, and United States all weighed in around 1,500!

Several could be rather bold and certainly many became quite happy with their profits derived from captured ships and good. Thomas Boyle, even notably proclaimed the entirety of the British Isles to be under blockade in a tongue in cheek response to the British blockade of the US which took quite some time to go from being in name only to actually effective.

But with the end of the war that marked the real end of American Privateers. Lack of any vastly superior maritime foe to defeat was part of it, and by the last decade of the 19th century the practice itself had fallen out of favor internationally while the US Navy was growing into a force that was large enough to both project power abroad and protect the sea lanes of communication.

It should be noted many were issued by the Confederate Government though their validity was of course challenged by the US and they experienced uneven recognition in European ports and courts. Something that was true of the Confederate Naval war effort in general.

The US issued Letters of Marque during the Quasi-War and the War of 1812. The Confederacy issued some during the Civil War.

The Quasi-War was an undeclared naval conflict between France and the US from 1798 to 1800. The US signed an alliance with France during the revolutionary war, including a mutual defense pact. However, the French monarchy was overthrown, and the US didn't want to be dragged into the subsequent British/French war. The US signed the Jay Treaty with Britain, affirming neutrality with the conflict. France didn't particularly like this state of affairs, and they sent privateers to prey on American shipping heading to Britain. The US tried to build up it's navy with the Naval Act of 1794, but the US was far behind their French rivals. The US returned the favor by commissioning 1,000 privateers to attack French ships.

During the War of 1812, the US navy was still pretty pathetic. President James Madison issued Letters of Marques during the war to attack British shipping to help supplement the US's meager fleet. Here is the letter issued to Captain Million: https://constitution.org/mil/lmr/1812amer1.htm

In 1856, most of the great powers of Europe signed the Treaty of Paris, which banned Letters of Marque. The US didn't that document. However, during the Civil War, Lincoln wanted to keep the European powers out of the war, and refused to issue letters of Marque to comply with the Treaty of Paris. The Confederacy didn't oblige the treaty and issued Letters of Marque to attack American vessels.

Since the Civil War, the US hasn't issued Letters of Marque. There is a legend that the Goodyear blimp had one during WW2, although that story been debunked. There was a recent small attempt to bring the Letters of Marque back. In 2007, Ron Paul introduced a bill into Congress that would allow the President to issue letters of Marque to take the property of Al Qaeda. However, so far attempts of revival have not stuck.