In a large bowl, add the vinegar, dijon, pesto, garlic, sugar, salt, and pepper. Whisk to combine. Add the chicken tenderloins, cover and chill for at least 2 hours, up to 6 hours.
Take the chicken out of the marinade and let come to room temperature, about 20 minutes. Add the remaining marinade to a small saucepan and bring to a boil, boil for 1 minute, take off the heat.
Heat the grill to high and prepare it for cooking. Lower the heat to medium, place the chicken directly on the grates, brush with the leftover marinate every 30 seconds or so. Turn occasionally until cooked through, about 3-5 minutes per side. They are ready when the internal temperature reaches 165°F. Make sure to use all the marinade up while the chicken is on the grill, this makes it extra flavorful!