Three compositions of high-entropy carbides, (TiHfVNbTa)C, (TiZrHfNbTa)C and (TiZrNbTaW)C were synthesized via a modified Pechini process, in which citric acid served as both a cation chelating agent and a carbon source. Through pyrolysis and spark plasma sintering, single phase high-entropy carbides were formed from the homogeneous precursors at a relatively low temperature of 1800 °C. The dispersion of cations in the polymer precursor facilitated shorter diffusion distances in polymer-derived materials, and thus compositional homogeneity was significantly improved relative to materials produced by a solid-state method, as quantified by a defined coefficient of variation applied to energy-dispersive X-ray spectroscopy. A finer microstructure in polymer-derived materials results in improved fracture toughness with a K_IC value of 4.29 MPa·m^1/2 achieved for (TiHfVNbTa)C.