Prepare the grill for direct heat and preheat to medium.
In a large bowl, whisk together the oil, vinegar, Italian seasoning, salt, and red pepper flakes. Add the vegetables and toss.
Spray the foil pieces with cooking spray. Divide the vegetables evenly among the foil packets. Drizzle any of the remaining dressing on top of the vegetables. Fold the long sides of foil toward each other, crimping edges to seal. Fold and crimp remaining edges, forming a sealed packet.
Place the foil packets on the grill, close the lid, and grill for 12-15 minutes. To check for doneness, remove one packet from the grill and unseal carefully. Taste to ensure the vegetables are crisp tender.
To serve, cut an X in each packet and top the veggies with the crumbled cheese and basil.