There have been 4 occasions when a country hosted the winter Olympics and did not win a gold medal.
France 1924
Switzerland 1928
Yugoslavia 1984
Canada 1988
For completeness, here are the host nations who did win olympic gold at home, and the gold count:
United States 1932 - 6
Germany 1936 - 3
Switzerland 1948 - 3
Norway 1952 - 7
Italy 1956 - 1
USA 1960 - 3
Austria 1964 - 4
France 1968 - 4
Japan 1972 - 1
Austria 1972 - 2
USA 1980 - 6
France 1992 - 3
Norway 1994 - 10
Japan 1998 - 5
USA 2002 - 10
Italy 2006 - 5
Canada 2010 - ??
As of 2010, the only country to never have won a gold medal in an Olympics it hosted is Yugoslavia. However, because this country no longer exists, there is technically no country that has never won a gold medal in an Olympics it hosted, after Canada won its first gold medal at the 2010 Winter Olympics in Vancouver.
No South American country has ever won a winter Olympic medal.
iceland
Through the 2010 Winter Games in Vancouver, no.
Ghana has never won a winter Olympic medal.
There were no Olympic Games in 2007. The summer games were in 2008 and the winter games were in 2010.
usa won the silver medal for vancover 2010 Olympic winter games
The US earned a total of 6 medals at the 1988 winter Olympic games.
China
212
Charles Jewtraw won the first ever Olympic winter gold medal in the 500 meter speed-skating.
Canada won both the men's and women's gold medal in ice hockey at the 2002 Winter Games in Salt Lake City.
No, Aruba has not yet won any Olympic medals in the summer or winter games.