Place chicken and mushrooms in separate gallon-size plastic zipper bags. Whisk together olive oil, vegetable broth, rosemary and pepper and pour half of mixture in each bag.
Seal bags and turn several times to coat food with rosemary oil. Bags may be prepared the night before and stored in refrigerator. If refrigerated, return chicken to room temperature before grilling.
Place chicken on grill skin side down and turn when skin turns deep brown. Grill chicken until crispy on outside and juices run clear when pierced with a fork, about 20 to 35 minutes. Remove and place on a platter; cover with foil and let rest for about 10 minutes.
While chicken cooks, thread mushrooms onto rosemary sprigs and other vegetables onto bamboo or metal skewers. Keep like vegetables together and brush them, excluding mushrooms, lightly with olive oil.
While chicken rests, grill vegetables. Start with firmer vegetables like mushrooms, onions and bell peppers, followed by softer ones like summer squash and tomatoes. Rotate once and grill until lightly charred and tender, about 10 minutes for firmer vegetables, 6 for softer vegetables.
Add vegetables to platter with chicken and serve.