In a medium bowl, whisk together the garlic, lemon juice, vinegar, oil, yogurt, and oregano. Transfer mixture to a large resealable bag and add the chicken. Refrigerate for about an hour, turning occasionally.
Preheat grill on medium-high heat. Remove chicken from marinade. Grill chicken until just cooked through, about 5 minutes per side, or until meat thermometer inserted into the chicken read 160 degrees F. Allow the chicken to rest for a few minutes before slicing into strips.
Wrap pitas in aluminium foil and place directly on oven rack and heat for 6-8 minutes.
Divide chicken strips evenly among pitas; top each gyro with 1/4 cup tzatziki sauce, tomato slices, red onion slices, and 2 tablespoons feta. Wrap aluminium foil around gyro and serve immediately.