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There are Emirates credit cards but, unfortunately, they are not offered to the U.S. residents. However, you can still find a credit card that will allow you to save on Emirates flights. For example, it can be any travel rewards credit card that is issued by a major bank and is not tied to a particular airline. Such a card will allow you to earn miles on flights and other everyday purchases. All earned miles can be redeemed for travel credits which include airfare, hotel stays, and rental cars.
For example, the Discover it® Miles card has all the best features of a travel card. You can earn 1.5x miles on all purchases, including Emirates. Earned miles can naturally be redeemed for all travel purchases, including Emirates. The card is pretty easy for use and extremely universal. Besides, there's the special feature of Discover. It matches all the miles you've earned by the end of the year (provided you are a new cardmember). There's no need to worry about where and when to spend this massive number of miles - miles never expire and you can take your time deciding whether to take an Emirates flight or another one. And finally, Discover doesn't have any annual fee. Good to know you aren't losing anything here.