A couple options:

If you want to do your API call before any processing of the submitted data takes place, use the 'NOTIFY_HEADER_START_CONTACT_US' hook. Do keep in mind that that hook point doesn't know whether the form has been submitted or not; it'll fire even when the contact page is merely being drawn, so you'll have to check for presence of POST data and do all your own validation before doing your API call.

Or you could fire on 'NOTIFY_EMAIL_AFTER_SEND_WITH_ALL_PARAMS' which receives an array of parameters based on the data used to prepare the email. You'll want to check the $module parameter and fire only when it's equal to 'contact_us'. The array is: array($to_name, $to_email_address, $from_email_name, $from_email_address, $email_subject, $email_html, $text, $module, $ErrorInfo))