The proof that a vertex algebra can lead to a Wightman QFT

You might find "An Introduction to Conformal Field Theory" by M. Gaberdiel (arXiv:hep-th/9910156v2) useful. He has a brief discussion of how in some cases chiral algebras can be assembled into Conformal Field Theories and has some further references. The language of this review is somewhere in-between the CFT language of physicists and the more formal VOA language of mathematicians.

I found Nikolov's paper "Vertex Algebras in Higher Dimensions and Globally Conformal Invariant Quantum Field Theory" arXiv:hep-th/0307235. The abstract:

We propose an extension of the definition of vertex algebras in arbitrary space-time dimensions together with their basic structure theory. An one-to-one correspondence between these vertex algebras and axiomatic quantum field theory (QFT) with global conformal invariance (GCI) is constructed.

Now he did not explicitly show how are these generalized vertex algebras related to the usual chiral vertex algebras. So in my master's thesis arXiv:1607.05078 I restricted myself to unitary (quasi-)vertex operator algebras and reversed Kac's proof.