If a payment is late, then your account is no longer considered to be "in good standing". Part of the requirement for being able to collect and use your rewards is that the account has to be in good standing. Disney VISA will take away those rewards immediately if a payment is late but they will restore them all once the account reverts back to being up to date. You do not have to pay the balance off in full, you only need to make the minimum payment on time in order to be considered to be in good standing.
To keep from ever having your rewards dollars disappear (and to avoid late fees), make the minimum payment online, by phone or thru your own bank when the bill arrives. You can still pay the remaining balance off in full when you write out the check and mail it later. That way, if you should mail the check too late or if it should somehow get delayed in the mail, you're still covered.