Yes, the drivers take credit cards for payment. They take Visa, Mastercard, Discover, American Express, Diners Club, Carte Blance, and JCB.
You will pay the driver that carries you from the airport to your resort the full roundtrip fee, plus HALF of the roundtrip tip when you arrive at your resort. You may use a credit card for this and write in the tip amount.
When you are picked up from your resort to return to the airport, you should pay your driver the other HALF of the tip in cash. (You might have a different driver).
Sometimes Tiffany contracts out to independent drivers, and these drivers are allowed to use their own merchant credit card systems if they have them. So when your credit card statement comes in and it shows a different transportation company on the credit card, don't be surprised.
I have also had a driver call me back weeks after my trip and ask if I could give him a Mastercard or Visa instead of the Diners Club card I used. I told him I would have to get back with him and immediately called Tiffany. They told me NOT to give him alternate payment, and to instruct him to stop by the Tiffany office to run the charge and they would reimburse him.
A few weeks later, the driver finally did.
I also understand that Tiffany requires cash for one-way trips, because the smaller amounts running through their merchant credit systems are not cost effective.
Carol