Heat gas or charcoal grill. Cut 4 (18x12-inch) sheets of heavy-duty foil. Place ham piece in center of each. Top each evenly with potatoes and vegetable mixture. Bring up foil sides so edges meet. Seal edges, making tight 1/2-inch fold; fold again, allowing space on sides for heat circulation and expansion.
Place packets, seam side up, on grill. Cover grill; cook over medium heat 18 to 23 minutes, rearranging packets several times during cooking, until potatoes and vegetables are tender.
To serve, carefully open packets to allow steam to escape. Sprinkle each with cheese; close packets and let stand until cheese is melted, 1 to 2 minutes.