私の環境では以下のソースで動いている。
sample.html
<!DOCTYPE html>
<html lang="ja">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta http-equiv="X-UA-Compatible" content="ie=edge">
<title>Document</title>
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/flatpickr/dist/flatpickr.min.css">
</head>
<body>
<input type="text" name="myDate" id="myDate">
<script src="https://npmcdn.com/flatpickr/dist/flatpickr.min.js"></script>
<script src="https://npmcdn.com/flatpickr/dist/l10n/ja.js"></script>
<script>
flatpickr("#myDate", {
"locale": "ja",
}
);
</script>
</body>
</html>