Heat olive oil in a large pot over medium heat; cook and stir onion, yellow bell pepper, and garlic in the hot oil until softened, 5 to 10 minutes. Add potatoes, carrots, zucchini, and enough water to cover mixture by 2 inches. Stir salsa into the mixture and add Italian seasoning, onion powder, garlic powder, black pepper, bay leaves, chili powder, and paprika.
Cook soup over medium-low, maintaining a slight boil, until potatoes soften, about 40 minutes. Add broccoli and peas; cook until broccoli is tender, about 20 more minutes. Remove bay leaves.