Zappa was born in Brescia, Italy. He began his professional career in the youth teams of local Lombardy club Lumezzane. In 2011, he was transferred to AlbinoLeffe, again in the youth sector. In 2013, he completed a transfer to Portuguese club S.C. Beira-Mar, before returning to Italy less than a year later to sign with Sassuolo for a fee of €220,000. After impressing for the youth sides, reigning Serie A champions Juventus made an approach for Zappa, initially signing him to a 6-month loan in February 2015. Zappa was also suspended for 45 days in 2015 due to the irregular transfer to Portugal. AlbinoLeffe also appealed to the Court of Arbitration for Sport in order to recover part of the transfer fee.