1.Is there any classification of symmetric structures on monoidal category $Vect$(with respect to usual tensor product)? 


2.Can one organize them in suitable category?


I hope that this is nice and instructive exercise, which is appropriate for mathoverflow.



Symmetric structures on Vect
There's just the one (up to symmetric monoidal equivalence). 


For this answer, I'll need a lemma: if $C$, $D$ are $Vect$-enriched categories and $F, G$ are additive $Vect$-enriched functors $C \to D$, then any natural transformation $F \to G$ is $Vect$-enriched natural. This is a consequence of the fact that the underlying-set functor $\hom(k, -): Vect \to Set$ (where $k$ is the ground field) is faithful; see Kelly's [Basic Concepts of Enriched Category Theory](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), bottom of page 10. 


Let $\otimes\_{bil}: Vect \times Vect \to Vect$ be the tensor product. (All tensor products are over the ground field.) The subscript $bil$ indicates that as it stands, the tensor product is not $Vect$-enriched, but at least it is "locally bilinear" in the sense that the structure maps 


$$\hom(V, V') \times \hom(W, W') \to hom(V \otimes W, V' \otimes W')$$ 


are bilinear. Similarly, let $\sigma\_{bil}: Vect \times Vect \to Vect \times Vect$ denote the obvious transposition functor, taking a pair $(V, W)$ to $(W, V)$. A symmetry structure on $Vect$ is, by definition, an invertible natural transformation of the form 


$$\otimes\_{bil} \to \otimes\_{bil} \circ \sigma\_{bil}$$ 


satisfying some extra coherence properties. 


There is an obvious passage from bilinearity to linearity, where we replace $Vect \times Vect$ by $Vect \otimes Vect$ (objects of $Vect \otimes Vect$ are again pairs $(V, W)$, but $\hom((V, V'), (W, W'))$ is the tensor product $\hom(V, V') \otimes \hom(W, W')$ instead of the cartesian product). We have corresponding functors $\otimes: Vect \otimes Vect \to Vect$, $\sigma: Vect \otimes Vect \to Vect \otimes Vect$, and the invertible natural transformations above are in natural bijection with invertible natural transformations of the form 


$$\otimes \to \otimes \circ \sigma$$ 


which are, by the lemma, $Vect$-enriched. 


In short, we are trying to understand $Vect$-enriched natural transformations of the form 


$$V \otimes W \to W \otimes V$$ 


or equivalently, extranatural transformations of the form 


$$V \to \hom(W, W \otimes V) \cong \hom(\hom(k, W), W \otimes V).$$ 


By the enriched Yoneda lemma, these are in natural bijection with enriched natural transformations of the form 


$$V \to k \otimes V \cong V.$$ 


Applying the Yoneda trick a second time, these are in natural correspondence with morphisms $k \to k$, i.e., scalars. Since the transformation $V \otimes W \to W \otimes V$ is to be invertible, the corresponding scalar $\lambda \in k$ is also invertible. 


Additionally, the required coherence condition puts another constraint on the scalar: if you work through the hexagon identity, you find out that $\lambda^2 = \lambda$. Therefore $\lambda = 1$, meaning that the only symmetry structure is the standard one. 


A key to the argument is the faithfulness of $\hom(k, -): Vect \to Set$. This gives a clue that to get more interesting examples where there is a variety of possible symmetric monoidal structures, you might pass to an enriched category context where the functor $hom(I, -): V \to Set$ represented by the monoidal unit $I$ is *not* faithful, as was the case with $V$ the category of $\mathbb{Z}\_2$-graded vector spaces, mentioned in an answer to another question of yours. 


