There was never a standing order for the U.S. Army to kill buffalo, but American commanders understood very well the connection between the buffalo and the Plains nations. In 1867, General Winfield Scott Hancock said this to a group of Arapaho chiefs:

You know well that the game is getting very scarce... cultivate the friendship of the white man, so that when the game is all gone, they may take care of you.

In May 1868, General William Tecumseh Sherman wrote to his subordinate, Gen. Philip Sheridan, that

as long as buffalo are up on the Republican [River] the Indians will go there. I think it would be wise to invite all the sportsmen of England and America there... Until the Buffalo and consequent Indians are out the Roads [the routes of the UP and Kansas Pacific railroads] we will have collisions and trouble.

A few months later, Sheridan wrote this to Sherman as he was planning to launch a campaign of reprisal against the Cheyenne for treaty violations, stating his plans were to:

make them poor by the destruction of their stock, and then settle them on the lands allotted to them.

Sheridan's commitment to the practice of total war is emphasized by his commands to Colonel George Custer in the winter 1868 campaign against the Cheyenne:

Destroy [Cheyenne] villages and ponies, kill or hang all warriors, and bring back all woman and children.

Even in times of peace, Sherman and Sheridan pursued their goal of slaughtering buffalo. They provided letters of introduction to private hunters, asking military posts to provide them with hospitality. In practice, this meant influential guests visiting the frontier would receive military escorts, guides, and even equipment and ammunition.

This policy was, of course, easily deniable if pursuing the deliberate starvation of the Plains nations provoked an outcry. In June 1869, the Army Navy Journal published a leaked remark of Sherman's:

the quickest way to compel the Indians to settle down to civilized life was to send ten regiments of soldiers to the plains, with orders to shoot buffaloes until they became too scarce to support the redskins.

This was widely read in government circles, and it can readily be interpreted as a quiet attempt by Sherman to gauge resistance to the idea. It provoked none. Sherman and the army pursued the extinction of the buffalo in every way but its open declaration. The provision of supplies to deniable private citizens to kill buffalo continued. Buffalo were hunted to supplement army rations, to keep herds away from the grazing pastures of cavalry horses, to sharpen the horseback shooting skills of soldiers, to practice the tracking skills needed on expeditions, and occasionally even to train artillery crews.

Sherman's enthusiasm was dampened as he gained familiarity with the Plains. He was frustrated by the apparent endlessness of the buffalo herds. Sherman and Sheridan believed there were hundreds of millions of buffalo on the Plains (although the true number was around 5 million, down from 60 million a century before); over the course of 1869 and 1870, the Army's quiet anti-buffalo campaign wound down. This satisfied many Army officers, who saw the campaign as a humiliating admission that the U.S. Army couldn't face Plains warriors on the field. The campaign's end was also influenced by President Grant's attempts to forge a lasting peace with the Plains nations, who were understandably threatened and infuriated by the wholesale slaughter of the buffalo herds.

This brief and semi-covert anti-buffalo campaign was eclipsed just a year later, however, with the 1871 invention of a process to make commercial leather out of bison hides - leather ideally suited to heavy machine belts.

The value of a buffalo hide skyrocketed and hunters swarmed to the Plains. The Army continued its policy of providing free ammunition and other equipment to the hunters. The buffalo herds were wiped out with amazing speed; the Plains wars of the 1870s, and the surrender of the last independent native nations, were a direct result.

In 1874 and 1875, Sherman and Sheridan waged a political campaign against a rising sentiment that the buffalo should be protected. In Texas, Sheridan spoke before the Texas legislature, saying:

For the sake of lasting peace, let them kill, skin, and sell until the buffaloes are exterminated.

Sherman personally convinced President Grant to pocket-veto a bill which would grant federal protection to the buffalo. The only concession was the maintenance of a miniscule herd, about 200 animals, in the newly established Yellowstone National Park.

In 1889, there were perhaps a thousand buffalo left in the world, with perhaps half of them still living in the wild.

Some historians argue the free market was responsible for this catastrophe, without any help from the Army, and they point to an 1879 letter from Sheridan to the Adjutant General:

The wholesale slaughter of the buffalo should be stopped.

This refers, though, only to hunters intruding on the Sioux Reservation and killing the small herd there, which kept the inhabitants from starving on inadequate government rations.

The actions of the U.S. Army and its commanders, regardless of the absence of written orders, speak volumes, especially given the strategic ideas expressed by Sherman and Sheridan in their correspondence. The Army killed vast numbers of buffalo, and evaded civilian oversight and treaty obligations by providing assistance to private citizens who completed the slaughter in a shockingly short time period.

References:

The Destruction of the Bison: An Environmental History, 1750–1920, by Andrew C. Isenberg.

Terrible Swift Sword: The Life of General Philip H. Sheridan, by Joseph Wheelan

/u/Prufrock451 provides good insight into the destruction of the buffalo. If you would like some further information, some of which is already covered here, I wrote a similar answer here.