Let $P_0,\ldots,P_n$ denote the coordinate hyperplanes in $\mathbb P^n$, and suppose that in each $P_i$ I have a degree $d$ hypersurface $V_i$. I am trying to understand what is the obstruction to the existence of a degree $d$ hypersurface $V \subset \mathbb P^n$ such that $V \cap P_i = V_i$.

There are some obvious obstructions: for example, my prescribed hyperplane sections $V_i$ must agree on intersections so that $V_i \cap P_j = V_j \cap P_i$. I don't think this alone is quite sufficient. In the case $n=3$ for example, let $p = P_1 \cap P_2 \cap P_3$ and suppose that $V_1,V_2,V_3$ all go through this point. Then $T_p V_1 + T_p V_2 + T_p V_3$ had better have dimension $2$, not $3$.

I am sure there is some exact sequence I'm missing - what is it?

(note: of course $V$ does not have to be unique, at least if $d \geq n+1$. You can add monomials containing all of the variables to the defining equation, and it doesn't change the intersection of the resulting surface with the coordinate hyperplanes)