In a food processor or blender, process all marinade ingredients except olive oil, and pulse until smooth. While motor is running, add olive oil in a thin stream until emulsified.
Slice thicker vegetables like squash and onions into ¼-inch thick slices. Seed and quarter peppers, and trim the bottoms from mushrooms. Place in a large bowl with venison. Pour marinade over, toss to coat, and refrigerate for 2 to 12 hours, depending on the cut of venison.
Remove venison and vegetables from marinade and drain well. Allow to rest at room temperature for 30 minutes before grilling.
Place venison and vegetables on a medium-hot grill until just cooked, but not overcooked.
Serve on individual plates or on a large platter "family style."
