admin_post hook not called

Modify your form to include:

<form action="<?php echo admin_url('admin-post.php'); ?>" method="post">
    <input type="hidden" name="action" value="volunteer">
    ...
</form>

and add an action to run your function:

add_action( 'admin_post_volunteer', 'send_contact_to_civicrm' );
add_action( 'admin_post_nopriv_volunteer', 'send_contact_to_civicrm' );

function send_contact_to_civicrm() {
   // stuff here
}