Vaessen was born in Gillingham to a footballing family – his father Leon had played for Millwall and Gillingham. A centre-forward, he joined Arsenal in 1977 as an apprentice, and made his debut aged 16 against Lokomotive Leipzig in the UEFA Cup on 27 September 1978. He made his league debut the following year, against Chelsea on 14 May 1979. He turned professional in July the same year, scoring five goals in 13 appearances in 1979–80