Instead of using the API, one can examine the Instagram userpage to get the id. Example code in PHP:
$html = file_get_contents("http://instagram.com/<username>");
$doc = new DOMDocument();
$doc->loadHTML($html);
$xpath = new DOMXPath($doc);
$js = $xpath->query('//body/script[@type="text/javascript"]')->item(1)->nodeValue;
$start = strpos($js, '{');
$end = strrpos($js, ';');
$json = substr($js, $start, $end - $start);
$data = json_decode($json, true);
$data = $data["entry_data"]["UserProfile"][0];
# The "userMedia" entry of $data now has the same structure as the "data" field
# in Instagram API responses to user endpoints queries
echo $data["user"]["id"];
Of course, this code has to be adapted if Instagram changes its page format.