Aston Oxborough played in goal for a young Norwich City Under-18s against Manchester City on 30th April 2013, which the Canaries lost 4-1. Most of their Under-18s were rested for the FA Youth Cup final. He had been part of Norwich's Under 13s squad in 2010/1. He was on the bench for Norwich's Under-18s against Middlesbrough on 13th April 2013

He was part of Norwich's Under-16 squad for the 2013/14 season

In January 2014, Aston spent time with an England training camp for goalkeepers. He made his England Under-16 debut in the 4-1 win over Denmark in February 2014, following in the footsteps of former Norwich youth team keepers Andy Marshall, Robert Green, Joe Lewis and Declan Rudd.

Aston signed on as an Academy Scholar with Norwich on 8th July 2014 and was part of the Under-18s side in 2014/15 and 2015/16.

On 3rd May 2016, it was announced that Aston had been offered a two year professional contract. He will be part of the development squad

He was on the first team bench three times during City's Championship winning campaign of 2018/19.

On 2nd August 2019 Oxborough joined National League South Wealdstone until January. He made his debut on 3rd August in a 4-1 home win over Dartford. He made a total of 19 appearances as Wealdstone won National League South.

In March 2020 he signed a new contract keeping him at Carrow Road until 2022.

In August 2020, Aston was linked with another loan spell away from Norwich with Salford City suggested as a destination. He actually went on trial with Salford during August 2020 but ruptured his Achilles tendon which saw the loan cancelled (in fact Salford signed Aston's former teammate Nick Hayes).

On 31st August 2021 he joined National League side Barnet on a season long loan.

He was released by Norwich on 20th May 2022.

In July 2022 it was reported that he was interesting Colchester United. He was also reported to be on trial with Motherwell by the Scottish Daily Express on 13th July 2022. This led to him signing a two year contract with Motherwell on 13th August 2022.