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We consider a scenario in which Sgr A* is in a massive black hole binary (MBHB) with an as-of-yet undetected supermassive or intermediate-mass black hole companion. Dynamical encounters between this MBHB and single stars in its immediate vicinity would eject hypervelocity stars (HVSs) with velocities beyond the Galactic escape velocity. In this work, we use existing HVS observations to constrain for the first time the existence of a companion to Sgr A*. We simulate the ejection of HVSs via the `MBHB slingshot' scenario and show that the population of HVSs detectable today depends strongly on the companion mass and the separation of the MBHB. We demonstrate that the lack of uncontroversial HVS candidates in \textit{Gaia} Data Release 3 places a firm upper limit on the mass of a possible Sgr A* companion. Within one milliparsec of Sgr A*, our results exclude a companion more massive than $1000 \, \mathrm{M_\odot}$. If Sgr A* recently merged with a companion black hole, our findings indicate that unless this companion was less massive than $500 \, \mathrm{M_\odot}$, this merger must have occurred at least $10$ Myr ago. These results complement and improve upon existing independent constraints on a companion to Sgr A* and show that large regions of its parameter space can now be ruled out.