[1][2] The New York Mets selected Orze in the fifth round, with the 150th overall pick, of the 2020 Major League Baseball draft.

[4] Orze made his professional debut in 2021 with the High–A Brooklyn Cyclones and was promoted to the Double–A Binghamton Rumble Ponies and Triple–A Syracuse Mets during the season.

[7] Orze returned to Syracuse in 2023, making 39 appearances and posting a 3–4 record and 5.31 ERA with 81 strikeouts over 61 innings.

[8] Orze began the 2024 campaign with Syracuse, compiling a 3.76 ERA with 56 strikeouts and two saves across 38+1⁄3 innings pitched.

On November 19, 2024, the Mets traded Orze to the Tampa Bay Rays in exchange for Jose Siri.