Season chicken breasts with salt, pepper, garlic powder, and ground ginger in a bowl.
Combine sweet chili sauce and soy sauce in a separate bowl; pour half over chicken and marinate in the refrigerator for at least 30 minutes.
Heat olive oil in a large skillet over medium heat and cook chicken for 6-7 minutes per side until fully cooked and internal temperature reaches 165°F.
Remove chicken from skillet and let it rest a few minutes before slicing into thin strips.
Warm tortillas in a microwave or dry skillet until pliable.
Assemble wraps by placing shredded lettuce, sliced chicken, shredded carrots, diced cucumbers, and cilantro on each tortilla.
Drizzle remaining sweet chili sauce over fillings, then fold sides and roll tortillas tightly.
Cut wraps in half diagonally and serve immediately with lime wedges.