Whisk vegetable oil, cider vinegar, garlic, sugar, and mustard together in a bowl to create dressing.
Place purple sweet potatoes, yam, and white sweet potato into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender but not soft, 5 to 7 minutes. Drain and rinse under cold water. Transfer to a large serving bowl.
Combine beans, corn, red bell pepper, yellow bell pepper, onion, and cilantro with the potatoes. Pour dressing over and toss to coat.