Oracle query’s required
Your problem is #P-hard. Indeed, given a #SAT instance with variables $x_i$ and clauses $C_j$, let $\kappa_{i,b}$ be the product of the clauses $C_j$ satisfied by the truth assignment $x_i=b$, and consider
$$ P = \prod_i (\kappa_{i,0} + \kappa_{i,1}). $$
The coefficient of $\prod_j C_j$ in $P$ is the number of satisfying assignments.

In the other direction, the special case where the input formula is $\Pi\Sigma\Pi$ (that is, the product of polynomials) is #P-complete. Suppose that we are interested in the coefficient of some monomial $m$ in $\prod_k P_k$, where the $P_k$ are polynomials. Substitute zero in all variables not appearing in $m$. Now guess one term from each $P_k$, and accept if the terms together cover $m$.