My solution. The final run is simplified using functions and logic in a while loop. I made a lot of notes too!

//handy functions
function print(message) {
var targetId = document.getElementById('output');
    targetId.innerHTML = message;}

function resetToStart(){
  searchResults = [];
  resultsFound = false;
}


//-----------------------------

//Place any global variables below

var searchKey;
var searchResults = [];
var resultsFound = false;
//-----------------------------

/* What are we trying to do?

User will 'lookup' the name of a student in the database. 
The print out will reflect the results of comparing the User's input with the data or a particular command.  

Scenarios will be 'record found' in which 'record presenter' will print a concatenated group of strings and pieces of data.

'quit' will be when the the word "quit" is typed in the prompt, or the User selects the cancel button.

If the User types something in the prompt that is not found in the record and not the string 'quit', then a message will print that '"User input" was not found.'
*/

function askQuestion(){   // Ask a question
searchKey = prompt('Enter a name to search the data base.  Type "quit" to quit the search');
}

function lookUp ( someInput ) { // Push any results into a new array called searchResults
    for (var i = 0 ; i < students.length; i += 1){
    var student = students[i];
    if (someInput.toLowerCase() === (student.name).toLowerCase()){
        searchResults.push(student)
    }
  }
}

function checkSearch (someArray) { // Were there any results in the array?  True or False?
  if (someArray.length > 0){
    resultsFound = true;
    } 
}

function recordPresenter(someArray) {
  var html = '';
  for ( i = 0; i < someArray.length; i += 1 ) {
    var record = someArray[i];
    html += '<h2>' + record.name + '</h2>';
    html += '<p>' + record.track + '<p>';
    html += '<p>' + record.achievements + '<p>';
    html += '<p>' + record.points + '<p>';
    print(html);
  }
}




// loop it and run it!

while (true) {
  resetToStart();
  askQuestion();

// use conditional here to quit the loop if we type 'quit' or if we select the cancel button

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


  lookUp(searchKey);
  checkSearch(searchResults);

  //--------------------------

  // User input logic

  if (resultsFound === true) {
  recordPresenter(searchResults);
  }   else if (resultsFound === false) {
      print('No records found');  
      }
}