Matt,
I have not done it with AJAX but if you are familiar with PHP you can use the cURL library
$access_token ="your token access token - this is obtained using 0auth";
$headers = array('Content-Type: application/json',sprintf('Authorization: Bearer %s', $access_token));
$ch = curl_init();
// set url
curl_setopt($ch, CURLOPT_URL, "https://<where your canvas is located >/api/v1/courses/:course_id/modules/:module_id/items/:id/done");
curl_setopt($ch, CURLOPT_HTTPHEADER, $headers);
//return the transfer as a string
curl_setopt($ch, CURLOPT_RETURNTRANSFER, 1);
curl_setopt($ch, CURLOPT_POST, true);
$output = curl_exec($ch);
// close curl resource to free up system resources
curl_close($ch);