If you pay your dues in full at the start of the year, Disney allows you to pay with a credit card. I guess they are willing to lose 2% to 3% to the credit card company in exchange for getting all the money for the whole year up front. If you prefer to "pay as you go" (monthly), then they apparently don't want to lose that 2% to 3% each month, so they won't charge the monthly payment to a credit card.
If money is tight at the start of the year, you can use autopay for the first month or two, then pay off the balance with a credit card to get at least some of the points, but you must pay the entire balance. You cannot make partial payments with a credit card.