Add the tomatoes to a food processor and pulse or puree until the desired texture is reached. Pour into a large bowl. Repeat the same process with the red peppers and cucumber, adding each ingredient to the same bowl.
Add the red onion and garlic to the food processor and pulse several times. Use a spatula to scrape down the sides of the bowl and pulse again. If a thinner/smoother texture is desired, pour some of the olive oil into the food processor while the machine is running, which will create a puree. Scrape the contents into the bowl with the other vegetables.
Stir in the remaining olive oil, tomato juice, and white wine vinegar. Season with salt and pepper to taste.
Cover the bowl and refrigerate overnight so the flavors have a chance to combine. Taste and adjust seasonings again if needed.
Serve chilled. Gazpacho keeps in the refrigerator for up to one week, but will taste best within 4-5 days.
Other tomato varieties may be substituted if heirlooms are unavailable. You can use a blender instead of a food processor if preferred.