If using wooden skewers, soak 8 in warm water for 30 minutes. Whisk garlic, mustard, salt, pepper, juice and vinegar. Slowly whisk in olive oil until emulsified. Stir in oregano.
Preheat a grill to medium-high. Place 2 tomatoes on each skewer, piercing crosswise, followed by a cube of Halloumi. Continue alternating tomatoes and Halloumi until each skewer has 5 or 6 tomatoes and 2 cubes of Halloumi.
Whisk vinaigrette and lightly brush onto skewers. Oil grill. Grill skewers, watching closely, until cheese is lightly browned and tomatoes are beginning to soften, about 3 minutes, turning halfway through. Transfer to a platter and drizzle with remaining vinaigrette. Serve immediately.