Organic salts represent an ideal experimental playground for studying the interplay between magnetic and charge degrees of freedom, which has culminated in the discovery of several spin-liquid candidates, such as $kappa$-(ET)$_2$Cu$_2$(CN)$_3$ ($kappa$-Cu). Recent theoretical studies indicate the possibility of chiral spin liquids stabilized by ring-exchange, but the parent states with chiral magnetic order have not been observed in this material family. In this work, we discuss the properties of the recently synthesized $kappa$-(BETS)$_2$Mn[N(CN)$_2$]$_3$ ($kappa$-Mn). Based on analysis of specific heat, magnetic torque, and NMR measurements combined with ab initio calculations, we identify a spin-vortex crystal order. These observations definitively confirm the importance of ring-exchange in these materials, and support the proposed chiral spin-liquid scenario for triangular lattice organics.