Vajna was born in Budapest, Hungary and emigrated to the US in 1956. His original profession was a hairdresser and wigmaker in Hong Kong. His first involvement in the film industry was the operation of movie theaters in the Far East. He founded a company with Mario Kassar called Carolco which went on to produce many blockbuster movies. He has done numerous films (frequently in association with Mario Kassar), including the Rambo movies, Die Hard with a Vengeance, and multiple Terminator movies and series.
On January 15, 2011, the Hungarian government appointed him government commissioner in charge of the Hungarian film industry. He is responsible for the strategy of the Hungarian Film preservation and development.
In 1989, he founded Intercom Ltd., Hungary's biggest movie and DVD distributor. He was also the CEO of the company.