In small saucepan, bring water to boil over high heat. Add carrots, return to boil. Reduce heat to medium-low. Cover and simmer 5 minutes. Drain. Move to separate bowl of ice water for 30 seconds to stop cooking.
In large resealable plastic food storage bag, add carrots, mushrooms, and bell peppers.
In small bowl, combine lemon juice, sugar, oil, garlic, fennel seeds, basil, and black pepper. Pour over vegetable mixture.
Close plastic bag securely, turn to coat. Place in refrigerator 8-24 hours, turning occasionally.
Drain, discard marinade, and place vegetables in serving dish.