Hi,
The $data PHP object contain only elements of the order, so it contains the payment methods.
In the preload part of the email we get all the payment method params, so if the expiry date is stored in the payment params you can get them from the PHP object $payment->payment_params.
But I am not sure that the expiry date will be stored there.
Else as Mohamed Thelji said, you can use a custom order field, set the expiry date there and that way you can display it in the email thanks to the code: $data->cart->customfield_name
So basically it require PHP knowledges and you have to know exactly how the expiry date is set.