The precise determination of satellite orbits is a complex problem that significantly affects quality of geodesy related applications, such as positioning, altimetry and gravity field determination. In this paper, the workflow and methodology adopted to develop a state-of-the-art open-source software, capable of estimating precise orbits of Low Earth Orbit (LEO) satellites using the Doppler Orbitography and Radiopositioning Integrated by Satellites (DORIS) system is presented. The tool features a modular architecture and implements advanced gravitational and non-gravitational force models consistent with current geodetic standards. DORIS measurements are ingested in RINEX format, and the corresponding observation equations are rigorously formulated and applied within the Precise Orbit Determination (POD) framework. A suitable high-accuracy numerical integrator and a sequential estimation algorithm have been implemented. Preliminary results using the Jason-3 satellite mission are compared against reference orbits from established analysis centers.