One solution is using Set data structure. A set is like an array, but contains no duplicated element.

function printOutVowels(str) {
  // Create an empty set
  const vowels = new Set();
  
  for (let i = 0; i < str.length; i++) {
    if (str[i] == "a" || str[i] == "e" || str[i] == "i" || str[i] == "o" || str[i] == "u") {
      // Add element to the set
      vowels.add(str[i]);
    }
  }

  // Print the set content
  for (let item of vowels) {
    console.log(item);
  }
}

printOutVowels("timidity");

Behind the scene, Set use its has method to check if an item is already in the set. This method is faster than Array.prototype.includes when a set and an array have the same size.

You could do that :

function printOutVowels(str){
    let vowels = "aeiou";
    let output = "";
    for (let i = 0; i < vowels.length; i++) {
      if(str.toLowerCase().includes(vowels[i])) output += vowels[i];
    }
    return output;
}

Splitting and filtering is a concise way to get the vowels. Running them through a set will enforce uniqueness. (Force to lowercase to handle both cases)

function vowels(str) {
  const isVowel = l => /^[aeiou]$/.test(l);
  const vowels = new Set(str.toLowerCase().split('').filter(isVowel));
  return Array.from(vowels.values());
}

console.log(vowels('TIMidity'))