Let's try a look at the 'machine gun type weapon operated by two or more soldiers', because handheld isn't going to work, and I see no reason to drag the poor horses into this mess. The problem with the Winans Steam Gun is that it's just markedly inferior to rifles and and the Gatling Gun. So let's discuss Gatling Guns.
The Gatling Gun, by which I mean the one invented by Richard J. Gatling, and not the modern guns which bear the classification 'gatling', was invented right before the Civil War, with the intention being to 'reduce the size of armies and show the futility of war', of which it only accomplished one of those things. It had a few unique features, like the 'rotating barrel' concept, letting it fire faster without the barrel overheating, and used a gravity fed hopper. Now, here's the interesting part - it was hand-cranked. And where I see a hand-crank, I see a spot for a steam engine.
In other words, if you hook up the steam engine to the gatling gun, you can eliminate the need for a person to hand-crank the gun and possibly improve the firing rate - though not too much, because at a certain point overheating starts being an issue, but you could work through those problem by increasing the number of barrels (it started at 6) and reducing the caliber of bullet (started at .58), and the increased weight wouldn't be an issue, because it's not hand cranked, perhaps even up to a thousand rounds a minute. (Possibly - the highest it actually got was 900.) At which point it's not exactly hand-held, and would take a team of soldiers to use, and is only slightly better than the original Gatling Gun, but does work. Slightly.
Though if you want, you can stretch plausibility slightly by mounting the thing on a traction engine, add some deflective armor, and you suddenly have an impromptu half-track slowly charging at the enemy. It won't work on uneven terrain, get stopped cold by trenches (which were everywhere), and be a massive target for enemy cannons but just imagine these things tearing through a Confederate charge.