My answer (including extended questions at the end)

// Lil Printer
function print(message) {
  var outputDiv = document.getElementById('output');
  outputDiv.innerHTML = message;
}
// Make an array to easily access student names w/ indices
// They are pushed as ".toUpperCase()" to make
// searches case insensitive
var studentNames = [];
for (var i = 0; i < students.length; i += 1) {
  studentNames.push(students[i].name.toUpperCase());
}
// This makes our report and then prints it via Lil Printer
// It will loop through the every entry of "recordsMatching" and use
// each value as the index in "students"
function getReport() {
  var report = '';
  for (var i = 0; i < recordsMatching.length; i += 1) {
    var recordIndex = recordsMatching[i];
    var student = students[recordIndex];
    report += '<h2>Student: ' + student.name + '</h2>';
    report += '<p>Track: ' + student.track + '</p>';
    report += '<p>Points: ' + student.points + '</p>';
    report += '<p>Achievements: ' + student.achievements + '</p>';
    print(report);
  }
}
// The Rest...
while(1) {
  var search = prompt('Enter the name of a student');
  // When and how to make the madness stop
    if (search === null || search.toLocaleUpperCase() === 'QUIT') {
    break;
  }
  // Make our searches case insensitive
  search = search.toUpperCase()
  // Look for a match to our search and store the index
  var idx = studentNames.indexOf(search);
  // If there is a match to our search
  if (idx > -1) {
    // Make an array to hold the indexes of our matches records
    var recordsMatching = [];
    // Push the index of each matching record to our new array
    while (idx != -1) {
      recordsMatching.push(idx);
      idx = studentNames.indexOf(search, idx+1);
    }
    getReport();
  }
  // If there isn't a match to our search
  else {
    print('<h2>That person isn\'t real\&hellip\; Sorry.');
  }
}