You can store the priorities in an array and then loop over the array and for every priority check if there's an object and whenever an object is found, return it. If the entire priorities array is exhausted then return null (or whatever you want).

const arr = [
    { name: "Bad", value: 2 },
    { name: "Critical", value: 1 },
    { name: "High", value: 5 },
    { name: "Medium", value: 5 },
  ],
  priorities = ["Critical", "High", "Medium", "Bad"],
  search = (arr, priorities) => {
    for (let p of priorities) {
      const obj = arr.find(({ name }) => name === p);
      if (obj) {
        return obj;
      }
    }
    return null;
  };

console.log(search(arr, priorities));

You can also sort the array based on the priority.

• Create a Map that stores the priorities.
• Sort arr based on the priorities stored in the map.

const arr = [
    { name: "Bad", value: 2 },
    { name: "Critical", value: 1 },
    { name: "High", value: 5 },
    { name: "Medium", value: 5 },
  ],
  priorities = new Map([
    ["Critical", 4],
    ["High", 3],
    ["Medium", 2],
    ["Bad", 1],
  ]),
  sortedArr = [...arr].sort(
    (a, b) => priorities.get(b.name) - priorities.get(a.name)
  );

console.log(sortedArr);

This will iterate over the array once until it finds the highest value and break the loop once it finds that value. Maximum number of iterations is N where N is x.length;

let x = [{name: 'Bad',value: 2},
{name: 'Critical', value: 1},
{name: 'High', value: 5},
{name: 'Medium', value: 5}]

function findVal(arr) {
  let returnVal = null;
  for (let i = 0; i < arr.length && returnVal === null; i++) {
    switch (arr[i].name) {
      case 'Critical':
        returnVal = arr[i];
        break;
      case 'High':
        returnVal = arr[i];
        break;
      case 'Medium':
        returnVal = arr[i];
        break;
      case 'Low':
        returnVal = arr[i];
        break;
      default:
        break;
    }
  }
  return returnVal;
}

findVal(x);

Note: It would be far simpler to designate a numerical value to the array values so that you could determine the severity simply by using the highest/lowest value rather than string matching

let value = x.find(x => x.name == "Critical") 
  || x.find(x => x.name == "High")
  || x.find(x => x.name == "Medium") ...