Preheat one side of grill to medium-high, the other side to medium-low.
In a nonstick saucepan over medium heat, add olive oil. Saute onions in oil until soft, about 2 minutes. Stir in vinegar, ketchup, brown sugar, Worcestershire, Dijon and a little salt and pepper. Bring sauce to a boil, reduce heat and simmer until reduced to 1½ cups, about 10 minutes. Set aside 1 cup sauce for serving.
Brush turkey tenderloins with ¼ cup remaining sauce. Grill turkey covered, on medium-low side of grill, 4 minutes. Flip turkey, baste with 2 tablespoons sauce, and grill 4 minutes. Baste again with 2 tablespoons sauce and transfer turkey to medium-high side. Cook basted turkey until an instant-read thermometer registers 165 degrees, about 4 minutes.
Remove turkey from grill and let rest for 5 minutes before slicing.
Drizzle the remaining sauce, that was set aside, over turkey slices and serve.