Start the coals or heat a gas grill for medium direct cooking. Make sure the grates are clean.
Whisk the oil, lime zest and juice, miso, and vinegar together in a small bowl until combined and thickened. Taste and adjust the seasoning with salt and pepper.
Trim the bottoms from the bok choy and cut into halves or quarters as needed. Pour half the vinaigrette into a large baking dish. Add the bok choy and turn in the vinaigrette until completely coated.
Put the bok choy on the grill directly over the fire. Close the lid and cook, turning once, until the leaves brown and you can insert a knife through the core with no resistance, 5 to 10 minutes per side, depending on their size. Transfer to a platter, drizzle with the reserved vinaigrette, and serve warm or at room temperature.