H-representation of convex hull
If you're okay with additional variables, then the answer is yes. This follows from Balas's extended formulation for the disjunction of polyhedra. It shows that $\operatorname{convex}.\operatorname{hull} ( \cup_j P_j)$ admits an extended formulation of size at most $\sum_{j}(\operatorname{xc}(P_j)+1)$, where $\operatorname{xc}(\cdot)$ denotes extension complexity. (The extension complexity of $P_j$ is at most the number of inequalities defining $P_j$.)

I have not thought about whether this can be done in the original space of variables.