I didn't want to comment, since it might take longer. I need more than the fact that through a general point of $X\subset \mathbb{P}^N$ of degree $n\leq N-2$, there is at least $N-n-1$ dimensional family of lines, but exactly of that dimension. Once we have that (and it is proved in Kollar's book, and this is where I need generality of the hypersurface), the subvariety $B\subset X$ of the union lines passing through this point is a complete intersection on $X$. So, the vector bundle $E$ on $X$ restricted to $B$ will be trivial mimicking the proof in Okonek et. al. Now, by boot strapping, since $B$ is a complete intersection on $X$, one can easily check that $E$ itself is trivial on $X$. If you need more details, please let me know.

I didn't want to comment, since it might take longer. I need more than the fact that through a general point of $X\subset \mathbb{P}^N$ of degree $n\leq N-2$, there is at least $N-n-1$ dimensional family of lines, but exactly of that dimension. Once we have that (and it is proved in Kollar's book, and this is where I need generality of the hypersurface), the subvariety $B\subset X$ of the union lines passing through this point is a complete intersection on $X$. So, the vector bundle $E$ on $X$ restricted to $B$ will be trivial mimicking the proof in Okonek et. al. $\dim B\geq 2$ by our assumption on $N,n$. So, $H^1(E|BB(k))=0$ for all $k$ since $E|B$ is trivial and $B$ is a complete intersection. Now, by boot strapping, since $B$ is a complete intersection on $X$, one can easily check that $E$ itself is trivial on $X$. If you need more details, please let me know.