Program Instant Pot for sauté. Add oil. When oil is hot, add garlic, onion, and both colors of bell pepper. Sauté 4 minutes.
Add salt, marinara, and chicken. Seal pot. Set valve to sealing. Program for manual-high and reduce time to 12 minutes.
Remove chicken from pot. Shred chicken with two forks. Add chicken back to instant pot along with mushrooms, zucchini, and tomatoes. Program for sauté. Sauté 5 minutes or until mushrooms are soft.
Warm flatbread in microwave 1 minute. Using a slotted spoon, arrange chicken mixture on flatbread. Sprinkle with basil ribbons.