How to make the submit button not appear on form unless an option is selected from radios?
You can use #states attribute in form api to hide submit button and check box

For Example

$form['email_canceled']['settings'] = array(
    '#type' =&gt; 'container',
    '#states' =&gt; array(

      // Hide the settings when the cancel notify checkbox is disabled.

      'invisible' =&gt; array(

       ':input[name="email_canceled_toggle"]' =&gt; array('checked' =&gt; FALSE),
      ),
    ),
  );