Grease the grill pan by brushing it with some oil. Preheat to medium-high heat.
Combine the mushrooms, onions, bell peppers and zucchini in a shallow dish. Whisk together the vinegar, salt, black pepper, oregano and garlic in a large bowl. Whisk in the 2 to 3 tablespoons oil to make the marinade. Add the vegetables to the marinade and let stand while the grill pan heats up.
Place the vegetables on the grill. Turn the vegetables once until they are all slightly charred and cooked through, about 20 minutes. Place the vegetables back into a large bowl with any leftover marinade. Toss and serve.