The actor, Harry Melling, has all his limbs. In order to play the role, there was some stage tricks, use of a motion base, and a lot of CGI. More about that can be read here.
In-story, how the character got this way is not addressed. Harry Melling believes it is a birth defect though:
You know, it’s important to to sort of have an idea of what happened to him. We’re talking post-Civil War, so having four limbs amputated is, from research, unheard of. Even having four limbs amputated now is a major major risk. So then you sort of steer towards the idea of this must be a something he had from birth and then you look into that line of research, and you find this particular syndrome that I sort of looked into. All of that work has to be done but not necessarily waved in front of the camera. How he moves is important. But all of that became secondary, because the story telling is this guy who went around performing these speeches and the audience is depleting.
Interview: Harry Melling talks The Ballad of Buster Scruggs, Netflix and Liam Neeson with horses