Vilmundur's father was Gylfi Þorsteinsson Gíslason, a member of parliament, government minister and professor.
Upon graduation, he studied in England and in 1971 he finished a BA Degree in literature and history at the University of Manchester.
Vilmundur was active in setting up radio programs on arts and culture, was a major participant in starting the Helgarpósturinn newspaper, and founded and served as editor of Nýtt land.
In the Icelandic parliament elections of 1971, Vilmundur stood as candidate for Social Democratic Party in the Westfjords constituency.
From October 1979 to February 1980 he served as Minister of Justice, Religion and Education in the minority Social Democratic government.