Free products of finite number of finite groups admit faithful actions by finite automaton permutations. Each action of this kind gives rise to Schreier graphs on levels and orbital Schreier graphs. These graphs depend on the action and a fixed generation set of the group. For the free product of two nontrivial cyclic finite groups we construct an orbital Schreier graph such that its underlying graph is planar. The proof relies on Kuratowski"s planarity criterion for infinite graphs and uses inductive limits of finite planar graphs.