As a child, Zampa bowled medium pace, but Cricket Australia had placed restrictions on how much pace bowlers were allowed to bowl in under-14 matches, so he decided to change his bowling style to leg spin, inspired by Australian Test cricketer Shane Warne. Zampa made his way into Australia's under-19 team in 2009 and represented Australia in their successful 2010 Under-19 Cricket World Cup campaign. Zampa played in two youth Test matches and eight youth ODIs, taking 11 wickets across the two formats.

In New South Wales (2010–2013)

As a result of Zampa's youth career for Australia, he was given a rookie contract with New South Wales in 2010, but he didn't get any opportunities to prove himself at state-level because New South Wales also had a number of other successful spin bowlers, such as Nathan Hauritz, Steve O'Keefe and Steve Smith. He did get another opportunity to represent Australia before his first-class debut, playing in the 2011 Hong Kong Cricket Sixes.

Zampa made his first-class debut for New South Wales in the 2012–13 Sheffield Shield season against Queensland. He took five wickets in a three-wicket win. Despite not having a contract with a Big Bash League team until a week before BBL|02 began, he played for the Sydney Thunder and was highly rated by Trevor Hohns. He finished the season having played three first-class matches, taking ten wickets at an average of 23 runs per wicket.

In South Australia (2013–2016)

After the season he accepted an offer to play for South Australia, in both first-class and one-day matches and for South Australia's Twenty20 team, the Adelaide Strikers. He was tempted by the guaranteed starts he would get in South Australia's team as they had just lost Test spinner Nathan Lyon to New South Wales. During the 2013 winter, he spent three months training at the Centre of Excellence before joining the South Australian squad for the 2013–14 season. The move to South Australia proved a useful one for progressing Zampa's career as he had the opportunity to work with experienced South African spinner Johan Botha, who was the captain of the South Australian side at the time of his move.

Zampa had an impressive performance in the 2014–15 Matador BBQs One-Day Cup, getting on a hat-trick and taking overall figures of 4/18 despite South Australia losing the match to Western Australia. In 2015, Zampa signed on for the Redbacks for two more seasons, but in order to get more experience, he changed Twenty20 sides to the Melbourne Stars, where he would have the opportunity to play alongside international stars Michael Clarke and Kevin Pietersen.