Module pattern in JavaScript

There are several problems with the posted code.

    var button = document.querySelectorAll("[class^='button']");
    button.onclick = this.writeEvent;

The result of querySelectorAll is a NodeList. Assigning to its onclick property will not achieve what you want. You want to assign to the onclick property of each individual node.

But actually that's not so simple, we'll need to come back to this.

  writeEvent: function(event){
    write(target.event)
  }

One problem here is that target is undefined. Surely you meant event.target.

Another problem is that write is also undefined. Perhaps you meant this.write, but that won't actually work well. The problem is that when writeEvent is called from a click event, it won't be called on the object, so this will not be bound to the calculator object, and the this.write call will raise an exception.

There's a way to overcome this, by binding the onclick handler function to the object when setting it.

Putting the above together:

var Calculator = {
  init: function() {
    var nodeList = document.querySelectorAll("[class^='button']");
    var callback = this.writeEvent.bind(this);
    nodeList.forEach(item => item.onclick = callback);
  },
  write: function(element) {
    document.getElementById("display").innerHTML = element;
  },
  writeEvent: function(event) {
    this.write(event.target);
  }
}

Calculator.init();