In a large, heavy pot, heat oil over medium-high heat. Add onions and cook for 5 minutes, stirring frequently. Reduce heat, add garlic, and cook 1 minute, stirring frequently. Add vinegar and bring to a boil. Stir in sugars and return to a boil, stirring constantly.
Heat four 1-cup jars and lids.
Add carrots, zucchini, rutabaga or radish, pepper, spice, and salt to the boiling mixture. Simmer gently for 35-40 minutes, stirring frequently, until liquid is thickened and vegetables are crisp-tender.
Fill hot jars with vegetables, sliding 1 bay leaf down the side of each jar. Pour hot brine atop vegetables, leaving 1/2-inch headspace. Top with lids. Process jars at a full rolling boil for 15 minutes.