Season chicken with salt and pepper.
In a large heavy sauté pan, heat olive oil over medium-high heat. Add chicken thighs to pan and sauté until just brown, approximately 5 minutes per side. If all 8 thighs don't fit in pan, sauté in 2 batches. Transfer chicken to plate and set aside.
Add bell pepper, onion, mushrooms, and garlic to pan. Sauté over medium heat until onion is tender, approximately 5 minutes. Season with salt and pepper.
Add ½ cup chicken stock and vinegar. Simmer until liquid is reduced by half, approximately 3 minutes.
Add tomatoes with juice, remaining stock, capers, olives, and oregano.
Return chicken to pan and coat in sauce. Simmer uncovered over medium-low heat until chicken is just cooked through, approximately 20 minutes.
Using tongs, transfer chicken to a platter. If necessary, boil sauce until slightly thickened, approximately 3 minutes. Spoon off any excess fat from atop sauce (see note). Spoon sauce over chicken, then sprinkle with basil and serve.