Quantum communication protocols seek to leverage the unique properties of quantum systems for coordination or communication tasks, usually with guarantees of security or anonymity that exceed what is possible classically. One promising domain of application is elections, where strong such guarantees are essential to ensure legitimacy. We experimentally implement a recently proposed election protocol from Centrone et al. [Quantum protocol for electronic voting without election authorities, Phys. Rev. Appl. 18, 014005 (2022)] designed such that no one, including a potential central authority, can know the preferred candidate of any voter other than themself. We conduct a four-party election, generating and distributing four-partite GHZ states with ≈89% fidelity and successfully recording voters’ intentions ≈87% of the time.