College Corinthians and Ireland Under-15 international Adam Idah played for Norwich's Under-16s in their 4-1 win over Ipswich in the Hospital Cup at Portman Road on 20th April 2017. Adam scored one of Norwich's goals with Jaden Thompson-Brissett scoring a hat-trick.

He played for Ireland in the UEFA Under-17 finals in May 2017 and was identified as being one of the players of the season.

On 21st July 2017, Norwich announced Adam's signing as a first year Academy Scholar. Norwich's Head of Academy Recruitment Gregg Broughton said “Idah is a centre-forward and was a regular playing up a year for Ireland U17s last season, scoring eight goals in 10 games. "

He played for Norwich in the Foyle Cup in Northern Ireland in July 2017, helping the Canaries reach the final before losing 1-0 to Kildrum.

On 30th June 2018, he signed his first professional contract with Norwich

Idah was called up to the Irish Under-21 squad in March 2019 and scored twice on his debut in a 3-0 win over Luxembourg on 24th March 2019. Idah played for Ireland in the Toulon Tournament in June 2019, reaching the semi-finals before losing to Brazil.

Idah made his Norwich debut in the 1-0 defeat to Crawley in the Carabao Cup (League Cup) on 27th August 2019.

After making his Premier League debut as a substitute against Crystal Palace on New Year's Day 2020, it was expected that Idah would go out on loan in the January 2020 transfer window with Doncaster heavily linked. Instead an injury crisis with Norwich's forward saw Idah picked to lead the line against Alex Neil's Preston in the FA Cup third round. Norwich won 4-2 at Deepdale with Idah scoring a hat-trick. His form hit the headlines and he was linked with a call-up to the full Ireland squad.

On 24th January 2020 it was reported that Charlton were keen on taking Idah on loan but Norwich manager Daniel Farke indicated that he wanted three forwards for the remainder of the 2019/20 season. He finished the season with 16 appearances and 3 goals as City were relegated to the Championship.

Idah made his full international debut for Ireland on 3rd September 2020, starting against Bulgaria in Sofia and playing 77 minutes of a 1-1 draw. New Ireland manager Stephen Kenny knew Idah from the Under-21s.

Idah scored his first league goal for Norwich in the 1-0 win at Huddersfield on 12th September 2020 on the opening day of the 2020/21 league campaign. He finished the 2020/21 campaign with 17 appearances (16 as substitute) and 3 goals as Norwich won the Championship title.

Injuries limited him to 22 appearances and 1 goal in 2021/22 as Norwich finished bottom of the Premier League.

On 31st October 2022 The Sun reported that Derby County manager Paul Warne is a big fan of Idah and would make a move to sign him on loan in January 2023.The report states that the Championship club will consider finding a loan destination for the young forward as he edges closer to getting back to full fitness. 

He made 27 appearances in Norwich's disappoining 2022/23 Championship season, scoring three goals.

On 27th January 2024 there were reports that Italian sides Sassuolo, Verona and Genoa were interested in signing Idah before the January 2024 transfer window closed. This was dismissed with Idah seen as a key player for Norwich with seven goals in 33 appearances. Verona made a loan to buy bid of 3.5million Euros but was rejected. Idah then signed on 1st February for Celtic on loan for the remainder of the 2023/24 season. "It's such a huge club with such a big history behind it and I can't even put into words what it's like to be at Celtic," said Idah. He made his Bhoys debut on 3rd February coming on as substitute and providing the assist in Celtic's 1-1 draw with Aberdeen.

Idah opened the scoring in Celtic's 5-0 win over Kilmarnock on 15th May 2024 that secured the Scottish Premiership title. Daizen Maeda, James Forrest, and a brace from Matt O'Riley secured Celtic's 54th league title. Idah then turned hero coming on as substitute to score the only goal of the game in the 90th minute of the Scottish FA Cup final against Rangers (who fielded Todd Cantwell in their line-up). This was his eighth goal in 18 games for Celtic. Idah said to BBC Scotland "I said I'd come here to try and score as many goals as I could, I didn't think I'd score a goal like this in front of all these fans. It's an amazing feeling, It's hard to put into words. Anyone here can see what it means to the fans, to the players. I've always supported Celtic, it's amazing. Football is tricky. I love it here, I've said it previously, it's a great club but I'm still contracted to Norwich. I'll go back in the summer and who knows?"

Celtic supposedly made an initial bid of £4.5million early in the summer which Norwich rejected. On 12th August 2024 it was reported that the two clubs had come to an agreement with a deal worth £8.5m, including guaranteed bonuses, plus a further £1m in potential add ons to take the Republic of Ireland striker back to Glasgow.City are also believed to have negotiated a 15pc sell on fee for Idah. The deal was completed on 14th August 2024. His last game for City was a substitute appearance in the 2-0 opening day defeat to Oxford. Idah said "Everyone back home, all my family and friends, they all wanted me to come back to Celtic, myself included, so it's massive and I'm just so happy to be here. Within the first couple of days of being here, when I came on loan first time, I wanted to play for Celtic. It was amazing to get all of those goals and to be in with the boys and to see these amazing fans week in, week out was unbelievable. I've never experienced anything like it and I'm just so happy to be back. I always wanted to play for Celtic and obviously last season was amazing."