Put the oil in a large skillet over medium-high heat. When it’s hot, add the chicken and sprinkle with salt and pepper. Cook until it’s browned and releases easily from the pan, 5 to 8 minutes. Turn and brown the other side. Transfer the chicken to a plate.
Pour off all but 2 tablespoons of the fat and return the pan to medium-high heat. Add the onion and celery and sprinkle with salt and pepper. Cook, stirring occasionally, until the vegetables soften, 3 to 5 minutes. Add the tomatoes, vinegar, and sugar and cook, scraping up any browned bits from the pan, until the sugar dissolves and the mixture boils, just a couple minutes.
Return the chicken to the pan and cook until the chicken is no longer pink inside and the sauce has thickened a bit, 10 to 15 minutes. Taste and adjust the seasoning, then serve the chicken hot, on the polenta if you like, with the sauce spooned over.