[3] Originally playing as an outfielder, Odumosu transitioned to a goalkeeper at under-12 level[4] before eventually signing for Malahide United.

Odumosu trialed in England with West Ham and Everton before signing for his hometown Dundalk as a youth player, where he played at U17 and U19 level.

[6] After trialing with Bolton Wanderers,[7] Odumosu signed for St Patrick's Athletic for the 2022 season to provide competition for West Ham United loanee Joseph Anang.

[8] Upon Anang returning to West Ham that summer, Odumosu would suffer an ankle injury leading to Clancy signing Danny Rogers.

[10] On 6 July 2023, Cliftonville announced the signing of David Odumosu on loan from St Patrick's Athletic to provide competition to first-choice keeper Nathan Gartside.