Assuming a gas giant is primarily gaseous past our visibility, is it possible that a moon traveling at a sufficient velocity would be able to exist within the atmosphere, or would terminal velocity from the atmosphere prevent an object of that size from maintaining appropriate centripetal force? For example, Neptune has winds that are so fast they move beyond the speed of sound. Would a moon be able to exist in an environment like that?
No, this can never be stable over long periods of time.
First of all, drag from the air in from of the moon would cause it to quickly deorbit and plunge into the gas giant. No amount of wind will be able to keep the moon up in the air. A gas giant's orbital speed near the cloud tops will be over 30 km/s (See the orbital velocity of Metis, Jupiter's innermost moon). There will be astronomical amounts of drag if the object is large. If the object is smaller, then it will be irregular, causing the same amount of drag. Also, a gas giant's weather is never constant. There will never be a circle in the gas giant's atmosphere that has a constant wind speed. These fluctuations will probably cause the moon to deorbit over time, or, if it is high in the atmosphere, to be thrown out of the atmosphere into the vacuum of space. The latter is highly unlikely.