Born in Hereford, Kieron had several youth clubs, including hometown Hereford United, however it was with Walsall that he came through ranks. Signing a professional deal in 2013, he made his senior debut for the Saddlers in a 3-1 League Cup defeat to Stoke City on 28th August that year.
Over the next two years, Morris would go out on loan, first to Conference North Worcester City. He made 8 appearances for the Victoria Ground side before returning to the Bescot Stadium.
In 2013-14, Morris had a brief loan period at Leamington, playing 5 times for the New Windmill Ground club, whilst on 30th January 2015, he joined Conference Premier side Wrexham until the end of the 2015-16 season. With the Dragons, Morris reached the 2015 FA Trophy final and made 13 appearances.
Returning to Walsall for 2015-16, Kieron established himself as a regular first team player, making 170 appearances and scoring 20 goals between August 2016 and January 2019.
Despite playing 26 games and scoring 5 goals at the start of 2018-19, mostly in League One, Morris was allowed to join Tranmere in League Two.
On 28th January 2019, Kieron arrived at Rovers on loan until the end of the 2018-19 season, joining Micky Mellon's squad for the remainder of the campaign. He made 17 appearances for Tranmere, including 82 minutes of Rovers' 2018-19 League Two Play-off Final victory over Newport County as the Super Whites returned to the third tier after a five-year absence.
On 26th June 2019, Kieron signed a 2-year contract with Rovers, making him a permanent member of the Super Whites.
Interesting Fact: Kieron played against Tranmere earlier in the 2018-19 season, representing Walsall in a 1-3 League Cup win for the Saddlers at Prenton Park on 14th August 2018.