On the hardness of constraint satisfaction
As pointed out by Antti Röyskö the solution $\mathbf{x} = 0$ always maximizes the objective function since  $\sum_{j=1}^n\mathbb{I}[\langle \mathbf{x}, \mathbf{v}_j\rangle \geq 0]$ corresponds to the number of constraints $\langle \mathbf{x}, \mathbf{v}_j\rangle \geq 0$ which are satsificed, and $\mathbf{x}$ satifies all constraints. 

However, the problem becomes much more interesting if we dont allow degenerate solutions. In fact, by removing $\mathbf{x} = 0$, the problem goes from having a closed form solution, to being NP-hard. It takes a bit of work, but you can actually reduce any integer feasibility program down to your problem.