When I saw this question, I thought I was in trouble, but I did come up with two places. One is in Frisco on Hatteras Island, the other is in Camden, NC, which is on the mainland.
Here is the link to the Driftwood Ranch in Frisco. The second is the Silver Star Farm located in Camden, NC. The second link will take you to Horseboarding.com, so if neither of these work, they may have a few more options for you.
One more thing, I am not sure where you will be staying or where you plan to ride on the Outer Banks, but if it is up in 4 Wheel Drive territory, where the Corolla wild horses are, I am not sure you will be permitted to do so. Issue with purity of the herd and could be dangerous with the wild horses about.