There is no office of Vietnam embassy in Neutral Zone up to now.
To request for legalization/verification of Vietnamese citizenship, etc., you have to contact the consulate or embassy of Vietnam in the nearest country to get detailed guidelines. You can find Vietnam embassy in the countries nearby Neutral Zone here.
To request for a Vietnamese visa, consider two options below:
Vietnam visa on arrival is also known as visa at airport and highly recommended for air travelers to Vietnam from Neutral Zone or any other country since it requires no contact with Vietnam embassy for application). See how Vietnam VOA works.