May a private citizen who owns an NFA firearm which is not registered have the firearm registered?

No. The NFA permits only manufacturers, makers, and importers to register firearms. Mere possessors may not register firearms. An unregistered NFA firearm is a contraband firearm,
and it is unlawful to possess, purchase, sell, or use the weapon. There are certain exceptions for items that were brought back from a war and where there is documentation showing that there was permission to bring the items back to the United States. While the ATF is under no obligation to do so, we have seen several cased where these items were presumed to have been registered and were put into the NFA registry. Otherwise, the possessor should contact the nearest ATF office to arrange for its disposition.

Violators may be fined not more than $250,000, and imprisoned not more than 10 years, or both. In addition, any vessel, vehicle or aircraft used to transport, conceal or possess an unregistered NFA firearm is subject to seizure and forfeiture, as is the weapon itself.

[49 U.S.C. 781-788, 26 U.S.C. 5861 and 5872]