No. On  $\mathbb P^1=\mathbb P^1(\mathbb C)$ we have $\Gamma(\mathbb P^1,\mathcal O_{\mathbb P^1}(-1))=\Gamma(\mathbb P^1,\mathcal O_{\mathbb P^1}(-2)=0$, but $O_{\mathbb P^1}(-1)$ and $O_{\mathbb P^1}(-2)$ are not isomorphic.  
   
However on an *affine algebraic* variety $X$, the answer is "yes". There is an amazing equivalence of categories between $\mathcal O(X)$-modules and the so-called quasi coherent sheaves on $X$. It is denoted  $M\mapsto \tilde M.$  
 In particular if you have a vector bundle $E$ on $X$, you can recover it (or rather its locally free associated sheaf)  from $M=\Gamma(X,)$ by this equivalence.   
And this remarkable result is not even very difficult! ( Hartshorne, *Algebraic Geometry*, II Corollary 5.5) . And it is valid on any affine *scheme*!