István Udvardi started playing water polo with Ferencvárosi TC in 1974 and was with them until 1981 and again in 1983-84. With Ferencvárosi, Udvardi won bronzes at the Hungarian championships in 1978, 1979, and 1984, Hungarian Cups in 1976-78, LEN Super Cup in 1978 and 1980, and LEN Cup Winners’ Cup in 1978 and 1980.
From 1981-82 Udvardi played with Szegedi EOL AK and then from 1985-87 was with Vasas SC, winning bronze at the 1986 Hungarian Championships. From 1988-92 Udvardi played in Italy, the first season with RN Savona and the next two seasons with RN Sori. Udvardi finished his career by playing one season in Turkey with Galatasaray. Udvardi was capped 52 times with the Hungarian national team from 1980-86. He won a 1980 Olympic bronze and a 1980 European Championships bronze.
Hungary