My first naive guess (in line with Andy's comment but not terribly well thought out) is that no, morphisms of VOAs are not physically terribly natural - rather the more natural thing to consider is an appropriate notion of bimodule for two VOAs, which would be domain walls between the corresponding chiral CFTs (or 3d TFTs, in the rational case).
To make an analogy one dimension down, let's think that we have an associative (or if you prefer $A_\infty$) algebra, and we use it to try to define a 2d TFT --- i.e. we can always integrate it over the circle to get a vector space (take Hochschild homology or center, depending on how you think of circles) and if it's fully dualizable (f.d. semisimple in the abelian case or "smooth proper" in the dg case) we can integrate it over 2-manifolds to get numbers. However from the TFT point of view what's important is the category of modules over the algebra rather than the algebra itself -- i.e., relations between two algebras are given by bimodules (ie Morita morphisms), not necessarily by morphisms of algebras. These are exactly domain walls between 2d field theories.
(This is also natural from thinking of 2d TFTs as noncommutative varieties --- only in the commutative case does it really make sense to focus on maps of algebras, since in that case you can recover the algebra from the corresponding category or TFT).
[If you want honest 2d CFTs rather than modular functors then you want a modification of the story above..]
Likewise I think (following Costello and Lurie) of a VOA as what you attach to a point in a 2d chiral CFT or modular functor (ie we're attaching vector spaces to Riemann surfaces, obtained by integrating the VOA over the surface --- conformal blocks, aka chiral homology). The coarse topological analog is an E_2 algebra. In any case what seems physically meaningful is domain walls between these modular functors (or the corresponding 3d TFT if it makes sense), and these are "chiral bimodules" for two vertex algebras: something you can put on a wall, so that on either side you have bulk operators given by your two VOAs (and in particular there's also a "boundary OPE" structure on this
chiral bimodule --- topologically this would be an associative algebra object in bimodules over two E_2 algebras).
Anyway to summarize usual morphisms of vertex algebras are special cases of something more natural, which are domain walls of chiral CFTs, which give monoidal functors between the monoidal categories of left modules ("boundary conditions") for the two VOAs..
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