An example of converting/formatting your date string without the use of Date or a library.
var monthLookup = {
  january: '01',
  february: '02',
  march: '03',
  april: '04',
  may: '05',
  june: '06',
  july: '07',
  august: '08',
  september: '09',
  october: '10',
  november: '11',
  december: '12'
};
function customParseAndformat(dateString) {
  var parts = dateString.split(' ');
  parts[0] = monthLookup[parts[0].toLowerCase()];
  parts[1] = parts[1] < 10 ? '0' + parts[1] : parts[1];
  return parts.join('/');
}
var myDateString = 'January 3 2016';
document.getElementById('out').textContent = customParseAndformat(myDateString);
<pre id="out"></pre>
 
 
You could use a similar method to parse and the create a Date object from the discrete values, which is far more reliable (cross-environment) than Date.parse. Example
var monthLookup = {
  january: '01',
  february: '02',
  march: '03',
  april: '04',
  may: '05',
  june: '06',
  july: '07',
  august: '08',
  september: '09',
  october: '10',
  november: '11',
  december: '12'
};
function parseAndCreateDate(dateString) {
  var parts = dateString.split(' ');
  parts[0] = monthLookup[parts[0].toLowerCase()] - 1;
  parts[1] = parts[1] < 10 ? '0' + parts[1] : parts[1];
  return new Date(parts[2], parts[0], parts[1]);
}
var myDateString = 'January 3 2016';
document.getElementById('out').textContent = parseAndCreateDate(myDateString).toDateString();
<pre id="out"></pre>
 
 
Now, you can format as you wish from the returned Date object, by using its methods.
Where can I find documentation on formatting a date in JavaScript?