Season chicken with salt and pepper. Heat a large cast-iron skillet over medium-high heat. Swirl in 1 tablespoon oil. Add chicken, skin-side down. Top with a parchment round; weight with another heavy skillet and canned goods. Cook until golden on skin side, about 5 minutes. Reduce heat to medium-low. Flip chicken; cook (with parchment and weighted skillet) until a thermometer inserted into thickest part of thigh (without touching bone) registers 165 degrees, 8 to 10 minutes. Transfer to a plate, reserving pan juices.
Return skillet to medium-high heat. Swirl in remaining 1 tablespoon oil. Add okra, beans, corn, bell pepper, and onion; season with salt and pepper. Cook, stirring, until vegetables are golden in spots, about 5 minutes. Remove from heat; stir in vinegar, then basil. Serve chicken with succotash, pan juices, and more basil leaves.