Lots of great flyfishing off the Outer Banks. Your options include sound side, ocean side, and offshore. There is somewhat of a divide north (Hattaras, etc.) vs. south of Ocracoke (Cape Lookout / Shackleford Banks
Let's start with the sound side. Earlier in the summer, you might encounter stripers migrating out to the ocean after their spawning migrations to the rivers. By mid-summer, you will encounter southern flounder (fluke) and speckled (spotted) trout and gray trout. Later, you will encounter red drum (aka puppy drum). Look for tidal channels and oyster reefs. You may also find smallish bluefish.
On the ocean side, you are likely to find more flounders and spotted trout in the troughs off the sand beaches and the inlets. In late summer/early fall, bigger bluefish and stripers can push 2-3lb spotted trout up on the beach. Especially from Ocracoke south, you can encounter schools of little tunny crashing baitfish either from the beach or just offshore in late sumer.
If y0u make it offshore, you can flyfish for mahi mahi off Sargassum mats in the Gulf Stream.