I also added an option named 'list' so the person who is looking for a student can see all the names available.

var dataStudent;
var finalReport = '';
var finalReportID = '';
var search;
var response;
var alertData;
var messageID = '';


function print(message) {
  var div = document.getElementById('output');
  div.innerHTML = message;
}

function getStudentData (datastudent) {
  var message = "<h2> ID: " + dataStudent.ID + "</h2>";
  message += "<p> Name: " + dataStudent.Name + "</p>";
  message += "<p> Proffession: " + dataStudent.Proffesion + "</p>";
  message += "<p> City: " + dataStudent.City + "</p>";
  message += "<p> Country: " + dataStudent.Country + "</p>";
  return message;
}

function getStudentDataID (datastudent) {
  messageID += "<h2> ID: " + dataStudent.ID + "</h2>";
  return messageID;
}


while (true) {
  search = prompt("Type the ID of the student or type 'quit' for exit. Type 'List' if you want to see all the ID numbers");
  response = parseInt(search);
  alertData = true;

  if (search.toLowerCase() === 'quit' || search === null) {
    break;
  } 
}

  for ( var i = 0; i < students.length; i += 1 ) {
    dataStudent = students[i];

    if (search.toLowerCase() === 'list') {
      finalReportID = getStudentDataID(dataStudent);
      print(finalReportID);  
    }
  }

  for ( var i = 0; i < students.length; i += 1 ) {
    idFound = students[i][0];
    dataStudent = students[i];
    if (dataStudent.ID === response) {
      alertData = false;
      finalReport = getStudentData(dataStudent);
      print(finalReport);
    } 
  }

  if (alertData && search.toLowerCase() !== 'list' ) {
      alert('Ups! This ID does not exist. Please, check the ID list and try again.');
    }
}