$\newcommand\Vect{\mathit{Vect}}\newcommand\Hom{\mathit{Hom}}$It seems that we can recover a smooth projective variety from its monoidal category of vector bundles using Cox rings of line bundles.

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.

To recover this information, let's discard all $R(X,L)$ that are not of finite type, and among the remaining ones consider those that are smooth projective varieties of maximal dimension. These all are birational to $X$ and we can recover $X$ as the only one of them (up to isomorphism) that admits regular maps to all the others.

$\newcommand\Vect{\mathit{Vect}}\newcommand\Hom{\mathit{Hom}}$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 $Y$.

It seems that we can recover a smooth projective variety from its monoidal category of vector bundles using Cox rings of line bundles.

For any line bundle $L$ on $X$ there is a dominant rational map from $X$ to $R(X,L):=\mathrm{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.

To recover this information, let's discard all $R(X,L)$ that are not of finite type, and among the remaining ones consider those that are smooth projective varieties of maximal dimension. These all are birational to $X$ and we can recover $X$ as the only one of them (up to isomorphism) that admits regular maps to all the others.

$\newcommand\Vect{\mathit{Vect}}\newcommand\Hom{\mathit{Hom}}$It seems that we can recover a smooth projective variety from its monoidal category of vector bundles using Cox rings of line bundles.

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.

To recover this information, let's discard all $R(X,L)$ that are not of finite type, and among the remaining ones consider those that are smooth projective varieties of maximal dimension. These all are birational to $X$ and we can recover $X$ as the only one of them (up to isomorphism) that admits regular maps to all the others.