What position would a casualty be placed in if they were vomiting a dark granulated substance?

Assuming no indication of spinal or head injuries, I'd use a coma or "side-stable" position if they were unconscious, or any position that allowed the head to turn so vomit would drain away and not be aspirated if conscious. I'd also consider raising the legs above the level of the heart to autotransfuse more blood to the core. Dark, granulated vomit is likely to be lysed blood from a gastrointestinal bleed (stomach or lower GI would be my guess, from the "coffee grounds" look of the blood vice something red or tarry). Id this is the case, they're very likely to vomit again, and aspiration of the vomitus can be extremely dangerous.

Put someone who cannot control their own vomiting on their side. The top leg would be placed over the bottom so they don't roll over.