Kamber spent his youth career at Basel and for a short time at Red Bull Salzburg. Kamber was a part of Basel's 2013–14 UEFA Youth League campaign, where he scored against Chelsea in a home match. In February 2015, he moved from Basel U21s to Servette and there, on 4 March 2015, he made his debut for the first team in the away game against Lausanne-Sport. He came on in the 64th minute for Alexandre Pasche. He scored his first goal for Servette in the 1–0 away win over Biel-Bienne, when he scored the winning goal shortly before the final whistle. At the end of the season, he came to a total of 14 appearances and three goals.
After his loan at Servette ended on 30 June 2015, Kamber joined Vaduz, signing a contract for three years until 30 June 2018. He made his debut on matchday 4 against FC Sion, where he came on as a substitute for Markus Neumayr. On 4 May 2016, he and his team won the Liechtenstein Football Cup, following a 11–0 final victory against Schaan. For the 2016–17 season, Kamber moved on loan to Winterthur in the Challenge League.
On 16 August 2022, Kamber moved to Poland to join Ekstraklasa side Górnik Zabrze on a two-year contract with an extension option.[3] He left the club by mutual consent on 25 March 2023.[4]
On 1 July 2023, Kamber signed with Schaffhausen on a free transfer.
