We study unconventional superconductivity in a two-dimensional locally noncentrosymmetric triangular lattice. The model is relevant to bilayer transition metal dichalcogenides with 2H(b) stacking structure, for example. The superconducting instability is analyzed by solving the linearized Eliashberg equation within the random phase approximation. We show that ferromagnetic fluctuations are dominant owing to the existence of disconnected Fermi pockets near van Hove singularity, and hence odd-parity spin-triplet superconductivity is favored. In the absence of the spin-orbit coupling, we find that odd-parity f-wave superconducting state is stabilized in a wide range of carrier density and interlayer coupling. Furthermore, we investigate impacts of the layer-dependent staggered Rashba and Zeeman spin-orbit coupling on the superconductivity. Multiple odd-parity superconducting phase diagrams are obtained as a function of the spin-orbit coupling and Coulomb interaction. Especially, a topological chiral p-wave pairing state is stabilized in the presence of a moderate Zeeman spin-orbit coupling. Our results shed light on a possibility of odd-parity superconductivity in various ferromagnetic van der Waals materials.