In this paper, we propose a way to realize topological s-wave superconductivity with the application of circularly polarized laser light in two-dimensional bilayer transition metal dichalcogenides (TMDs). Using Floquet theory, we analyze a tight-binding model of bilayer TMDs with time-periodic electric fields. After deriving an effective Hamiltonian, we investigate topological properties of the s-wave superconducting state. The laser light induces valley-dependent layer polarization and makes the system a topologically nontrivial superconducting state characterized by the Chern number. We show topological phase diagrams in the absence and presence of the Kane-Mele spin-orbit coupling which causes hidden spin polarization in bilayer TMDs. Although the topological phase diagram is affected by the spin-orbit coupling, topological superconductivity can be realized without relying on the spin-orbit coupling in sharp contrast to a previous proposal of laser-induced topological superconductivity [Takasan et al., Phys. Rev. B 95. 134508 (2017)]. We also discuss experimental setups to detect the topological phase.