Preheat a gas grill on high, covered, for 10 minutes, and then lower heat to medium. On a lightly floured work surface, press and stretch each piece of dough into a 9-inch round. Place on lightly floured baking sheets; set aside.
Place zucchini, onion, pepper and eggplant on a rimmed baking sheet. Brush with 2 Tbsp. olive oil and sprinkle with salt and pepper. Grill until vegetables are soft, turning once, 6 to 8 minutes total. Transfer to baking sheet and set aside.
Brush tops of dough rounds with 1 Tbsp. olive oil and carefully flip them, oiled side down, onto grill. Grill, uncovered, until undersides are golden, 2 to 3 minutes. Use tongs to place pizzas back on baking sheets. Brush uncooked sides with remaining 1 Tbsp. oil and flip so grilled sides of pizzas are facing up. After grilling second side, remove rounds and spread 1/4 cup tomato sauce on each pizza. Arrange vegetables on top of sauce and sprinkle with cheese. Slide pizzas back onto grill and cook, covered, until cheese is melted, 3 minutes.