The notion of vertex algebra, like any reasonable algebraic notion, makes sense inside any (sufficiently linear) symmetric monoidal category. The standard pictures of the operator product, however, suggest that symmetry is more than is needed: one should be able to formulate the axioms of "vertex algebra" internal to any (sufficiently linear) braided monoidal category. Has this been done in the literature? Are there standard examples?
For the case of vector spaces graded by an abelian group (with braiding determined by an abelian 3-cocycle following Joyal-Street), this was done by Dong and Lepowsky in their 1993 book "Generalized Vertex Algebras and Relative Vertex Operators". The object has the name "abelian intertwining algebra", and standard examples come from applying the lattice vertex algebra construction to rational lattices. A greater supply of examples was given in van Ekeren, Möller, Scheithauer by an orbifold construction in 2015.
As far as I can tell, the general definition has not appeared explicitly in the literature. Some variants have appeared under the name "quantum vertex algebra", though. In particular, I remember working out that Borcherds's definition of (A,H,S)-vertex algebra in his "Quantum Vertex Algebras" paper can be modified to fit the braided setting without too much trouble.