Combine chicken broth, soy sauce, sesame oil, rice wine vinegar, lemon juice, garlic, chile-garlic sauce, and salt in a large pot over medium heat; bring to a simmer and cook until hot, about 10 minutes. While the broth simmers, bring water to a boil in a separate pot. Cook wontons in boiling water until heated through, about 5 minutes. Remove from water with a slotted spoon and add to the soup.