Mix Italian dressing, yellow squash, zucchini, cherry tomatoes, and spring onions together in a large bowl. Cover with plastic wrap; let marinate in the refrigerator for 2 hours.
Place a vegetable tray in a smoker and heat to 250 degrees F (121 degrees C) according to manufacturer's instructions. Add wood chips.
Transfer marinated vegetables to the tray, reserving excess dressing in the bowl.
Cook vegetables, basting every 20 minutes with 1/3 of the reserved dressing, until tender, about 1 hour.