Hi,
From what I can see, it comes from a customization of the contact form in your template.
If you switch temporarily to the default template of Joomla and try again, I suppose that you see the captcha, right ?
In that case, remove the code:
if($option=='com_contact' && version_compare(JVERSION,'3.0','>=')){
$return = '<div class="control-group"><div class="control-label"><label id="recaptcha_response_field-lbl" for="recaptcha_response_field" class="required">'.$subText.'<span class="star"> *</span></label></div>';
$return .= '<div class="controls">'.$c->recaptcha_get_html($this->params->get('public_key'),$this->params->get('theme')).'</div></div>';
$replace = 'id="'.$id.'".{0,'.$components[$option]['lengthafter'].'}(<\/div>\r?\n?\t*<\/div>)';
}else
from the file plugins/system/recaptcha2/recaptcha2.php and that should help with the customized contact form.