Preheat grill with lid closed to high temperature, around 600 °F.
While grill is heating for pizza, grill the onions and peppers. Brush each cut side of the onion with vegetable oil. Toss the pepper rings with a small amount of vegetable oil, just to lightly coat, about 1-2 teaspoons. Lay onion, cut side down, on the heated grill and grill covered for about 5 minutes. Also lay the pepper rings on the grill and grill covered for about 1 minute per side. (The goal is not to fully cooked veggies, just give a little grill flavor and add grill marks.) When veggies are done, remove from grill, close lid and continue heating grill. Carefully slice the onion into rings.
Assemble the pizza: Generously flour a large flat cutting board or rimless baking sheet (preferred). Stretch or roll pizza dough to desired thickness and lay on board or baking sheet. Drizzle olive oil on dough and spread evenly to all the edges of the dough. Sprinkle oiled dough with cracked pepper and Italian seasoning and gently press the seasoning into the dough with your hands.
Layer the tomato slices, peppers and onions on pizza as desired. Gently press the veggies into the dough to help them stay in place. Sprinkle goat cheese on top of pizza.
Grill pizza: Turn off one burner of preheated grill. Oil grill grates above the off burner. Slide pizza off pan and onto oiled grill. Close lid quickly and cook for about 4 minutes.
Open lid and use a pizza peel or large grilling spatula to turn the pizza 180° on the grill. Close lid and continue grill for about 4-5 minutes, until edges are golden brown. Total cook time will vary based on grill and temperature.
Use pizza peel or large spatula to slide pizza off the grill and back onto baking sheet. Allow to cool for about 5 minutes and slice into wedges. Enjoy!