In what year was asbestos stopped being used in linoleum/vinyl
flooring for new construction, single family home in Wake County,
North Carolina?

Here is a site that indicates that asbestos was used in linoleum until 1970.
http://www.improvenet.com/adviceandresources/articleslibrary/jeff_williams_linoleum.html

This site indicates that asbestos was used in vinyl tiling at least
until 1980, but also tells where you can have such tested.
http://www.asbestosresource.com/asbestos/tile.html?print=true

It is, of course, possible that a builder used such materials after
the date when they ceased to be produced.
The scare about asbestos is in my humble opinion entirely overblown.
Yes, unfortunately people working in plants using asbestos have
suffered, but not people living in buildings with the stuff.
If you remove it, correct disposal could be costly.