Use an event-listener to trigger the submit and the clearing.
First, change the submit button to a regular button with a proper id (you should do the same to the other elements):
<input type="button" name="submitButton" id="submitButton" value="Submit" />
Then bind the event-listener to the button with JavaScript:
<script type="text/javascript">
    document.getElementById('submitButton').addEventListener('click', function ()
    {
        handleTheForm;
    }, false);
</script>
Wherea handleTheform is a method, defined accordingly:
<script type="text/javascript">
    function handleTheForm()
    {
        document.forms[0].submit(); // Submit the form.
        document.forms[0].reset(); // Clear the form.
    }
</script>
Edit To handle the Enter button, simply add an event-listener for buttons and check which key is being pressed:
<script type="text/javascript">
    document.addEventListener('keypress', function (e)
    {
        var key = e.which || e.keyCode;
        var enterKey = 13;
        if (key == enterKey)
        {
            handleTheForm;
        }
    });
</script>
Your final picture should look something like this:
<form id="myform" action="https://example.com" target="myiframe" method="POST">
  <input type="text" name="email" id="email" value="" />
  <input type="text" name="name" id="name" value="" />
  <input type="text" name="phone" id="phone" value="" />
  <input type="button" name="submitButton" id="submitButton" value="Submit" />
</form>
<script type="text/javascript">
    function handleTheForm()
    {
        document.forms[0].submit(); // Submit the form.
        document.forms[0].reset(); // Clear the form.
    }
    document.getElementById('submitButton').addEventListener('click', function ()
    {
        handleTheForm;
    }, false);
    document.addEventListener('keypress', function (e)
    {
        var key = e.which || e.keyCode;
        var enterKey = 13;
        if (key == enterKey)
        {
            handleTheForm;
        }
    });
</script>
You might have to tweek something a little but since I haven't manage to test this.