Reconstruct a variety from the category of locally free sheaves We know that the category of the (quasi-)coherent sheaves on a smooth projective variety $X$ determine the variety (aka. Gabriel–Rosenberg reconstruction theorem) and the derived category of coherent sheaves on a smooth projective variety $X$ with $\omega_X$ ample or anti-ample determine the variety (aka. Bondal–Orlov reconstruction theorem).
Do we have any known results about whether the category of algebraic vector bundles (or locally free sheaves) $\mathbf{Vect}_a(X)$ can determine a variety?
$$\mathbf{Vect}_a(X)\cong \mathbf{Vect}_a(Y)\implies X\cong Y$$
Does it work when the equivalence is between categories, $k$-linear categories or a $k$-linear tensor categories?
Edit: refer to Martin Brandenburg in the comment, the affine case is algebraic and one can show it by corresponding results in algebra.
 A: $\newcommand\Vect{\mathit{Vect}}\newcommand\Hom{\mathit{Hom}}$At least the birational tyie of a smooth projective variety can be recovered from the monoidal category of vector bundles on it. (the previous version of this answer claimed that the isomorphism class can be recovered but now I don't think that the original argument works)
For any line bundle $L$ on $X$ there is a dominant rational map from $X$ to $R(X,L):=\operatorname{Proj}\bigoplus H^0(X,L^n) $ which is an isomorphism if $L$ is very ample. In the monoidal category $\Vect(X)$ the invertible objects are precisely the line bundles, and the structure sheaf is the unit object, so given the category $\Vect(X)$ we can recover the collection of schemes $R(X,L)$ for all line bundles $L$ (because $H^0(X,L^n)=\Hom_{\Vect(X)}(O_X,L^{\otimes n})$), though we are not being told which of these arise from ample line bundles.
Let's discard all $R(X,L)$ that are not of finite type. Among the remaining ones all the $R(X,L)$s that have maximal dimension are birational to $X$, so the birational type of $X$ can be recovered from $(Vect(X),\otimes)$.
It is unclear to me right now how to recover $X$ itself: the issue is that an equivalence $(Vect(X),\otimes)\simeq (Vect(Y),\otimes)$ might a priori carry an ample line bundle on $X$ to a non-ample (though necessarily big) line bundle on another variety $Y$.
