Given an very ample line bundle $L$ on a projective variety we embed it into a projective space such that pullback of $\mathcal{O}(1)$ is $L$. Then we can identify the two. So in the category of vector bundles of the form $\bigoplus \mathcal{O}(n_i)$ where $n_i\geq 0$, obviously we can lift each object from the variety to the ambient projective space. Now consider a projective variety $X$ a subvariety $Z$ (assume $Z$ is a hyperplane section). Is it true that we can lift the morphisms between two objects in the form $\bigoplus \mathcal{O}_Z(n_i)$ to a neighborhood of $Z$ in $X$?(or formal completion of $Z$ along $X$). If so is it possible to give a functorial lift from the category of vector bundles of the aforementioned type on $Z$ to the formal completion $X_Z$? (or some neighborhood of $Z$ along $X$)

If these are not true in general I'd like to know cases that they can be true.