Q:

How to submit iron-form and trigger form's action?

I have this form that is supposed to post the input values to the formspree API using the form's action attribute on submit. After I added is="iron-form", it just posts the values out but doesn't trigger the action attribute for some reason.
<form is="iron-form" action="https://formspree.io/MY_EMAIL_HERE" method="post">

  <paper-input type="email" name="email" label="Your email" required></paper-input>    
  <paper-textarea name="message" label="Your message" required char-counter maxlength="2000"></paper-textarea>

  <input type="hidden" name="_next" value="{{baseUrl}}contact_form_sent" />

  <paper-button class="sendButton" on-click="_submit">Send</paper-button>
</form>

...

<script>
  Polymer({
    is: 'my-contact',
    _submit: function() {
      this.$.form.submit();
    }
  });
</script>

A:

this.$ is a map of IDs to nodes in the current Polymer element (see Polymer's automatic node finding). So, this.$.form is trying to access the element with an ID of "form", but there are no elements with that ID, so the value would be undefined.
You can fix this by giving your <form> an ID of "form":
<form id="form" ...>

HTMLImports.whenReady(() => {
  Polymer({
    is: 'x-foo',
    _submit: function() {
      this.$.form.submit();
    },
    _onResponse: function(e) {
      this._response = JSON.stringify(e.detail.response, null, 2);
    }
  });
});
<head>
  <base href="https://polygit.org/polymer+1.8.0/components/">
  <script src="webcomponentsjs/webcomponents-lite.min.js"></script>
  <link rel="import" href="polymer/polymer.html">
  <link rel="import" href="paper-button/paper-button.html">
  <link rel="import" href="iron-form/iron-form.html">
</head>
<body>
  <x-foo></x-foo>

  <dom-module id="x-foo">
    <template>
      <form id="form"
            is="iron-form"
            action="//httpbin.org/post"
            on-iron-form-response="_onResponse"
            method="post">
        <input type="text" name="test" value="foo"/>
        <paper-button raised on-tap="_submit">Submit</paper-button>
      </form>
      <pre>[[_response]]</pre>
    </template>
  </dom-module>
</body>

codepen

