What is the closest the United States has gotten to a military coup?

Probably Shays' Rebellion of 1786-1787. Militia soldier and Revolutionary War veteran Daniel Shays led 4000 troops to take over the Springfield (Massachusetts) Armory, which held a large stockpile of the U.S. Army's weaponry. The plan was to use these weapons to overthrow the still new United States government.

However, the small militia defending the Armory under the command of William Shepherd was able to hold off Shays' forces long enough that reinforcements arrived from Worcester under the command of former Continental Army general Benjamin Lincoln. The rebellion was put down, a couple of the leaders were executed, though Shays himself escaped to Vermont (then an independent country). He was pardoned and was allowed to return to Massachusetts, but his public image had suffered greatly and no further attempt by him to overthrow the government was ever made.

Sources:

The Story of Massachusetts by Edward Everett Hale

The History of the Insurrections, in Massachusetts, in the year 1786, and the Rebellion Consequent Thereon

Further reading:

In Debt to Shays: The Bicentennial of an Agrarian Rebellion

The Shays Rebellion: A Political Aftermath

A Few Notes on the Shays Rebellion

Shays's Rebellion: The American Revolution's Final Battle

EDIT:

You may also be interested in Leisler's Rebellion and the Boston Revolt of 1689, though that was pre-United States. The governor of the "Dominion of New England" which at that time included New Jersey, New York, Massachusetts, Connecticut, and Rhode Island, was a guy named Sir Edmund Andros. His rule was very unpopular, and an uprising was organized in Boston that eventually forced him into house arrest.

Sympathizers to this revolt down in New York City under the leadership of militia officer Jacob Leisler took over Fort James (a.k.a. Fort Amsterdam) which held the treasury for the government of the New York colony. After a few months, the British Navy arrived and broke up the rebellion peacefully, though Leisler and his son-in-law were controversially executed--controversial because Leisler's takeover of the fort had been accomplished without firing any shots and nobody else was killed during the rebellion. For the next couple decades, New York politics was organized very much along the lines of pro-Leisler and anti-Leisler factions, which largely overlapped along the ethnic lines of Dutch and English colonists of New York.

As for Andros, he was recalled to Britain. The Dominion of New England was dissolved, the British government re-organized their colonial administration, and the colony of Massachusetts was given a new charter. A new governor was appointed there, which placated the population and prevented any further unrest.

Sources:

Colonial New York: A History by Michale Kammen

The Story of Massachusetts by Edward Everett Hale