We give an equivalence of categories between: (i) M\"obius vertex algebras which are equipped with a choice of generating family of quasiprimary vectors, and (ii) (not-necessarily-unitary) M\"obius-covariant Wightman conformal field theories on the unit circle. We do not impose any technical restrictions on the theories considered (such as finite-dimensional conformal weight spaces or simplicity), yielding the most general equivalence between these two axiomatizations of two-dimensional chiral conformal field theory. This provides new opportunities to study non-unitary vertex algebras using the lens of algebraic conformal field theory and operator algebras, which we demonstrate by establishing a non-unitary version of the Reeh-Schlieder theorem.