Судя по документации примерно так (если без cURL):
$context = stream_context_create(array('http' => array(
'method' => 'POST',
'header' => "Content-Type: application/x-www-form-urlencoded\r\n".
"Authorization: Basic ".base64_encode("<app key>:<app secret>")."\r\n",
'content' => http_build_query(array(
'code' => <authorization code>,
'grant_type' => 'authorization_code',
'redirect_uri' => <redirect URI>,
)),
)));
$url = 'https://api.dropbox.com/1/oauth2/token';
$result = file_get_contents($url, false, $context);
$result_json = json_decode($result);
echo $result_json->access_token;
SDK у них хорошо расписано, бери да пользуйся.