This study proposes a topology optimization framework for transient reaction-diffusion problems coupled with finite elasticity. Unlike steady-state approaches, our formulation explicitly considers the time-dependent evolution of chemical concentrations and the resulting swelling-induced large deformations. In this coupled system, the local concentration of chemical species drives volumetric expansion via an eigenstrain mechanism, while the finite deformation simultaneously alters the diffusion pathways and reaction kinetics, creating a complex two-way interaction. The material distribution is parameterized using a density-based approach, where the optimization problem is formulated to maximize the mechanical output. We derive a generalized sensitivity analysis scheme based on the continuous adjoint method. This rigorous derivation involves the backward-in-time integration of the adjoint variables to accurately capture the history-dependent sensitivity of the transient phenomena, ensuring the gradient information accounts for the full temporal evolution. The proposed framework is applied to the design of self-oscillating Belousov-Zhabotinsky (BZ) gel structures, where chemical reactions induce periodic volumetric changes. Numerical examples demonstrate that the method successfully generates optimal material distributions that function as autonomous soft actuators, effectively translating internal chemical energy into desired mechanical motion without external electrical controls. These results validate the effectiveness of the proposed approach for designing dynamic functional materials involving complex multiphysics coupling. Beyond hydrogels, the generality of this framework suggests broader applicability to other systems governed by coupled diffusion and large deformation, such as the stress-diffusion coupling in battery electrodes and the stimuli-responsive deformation of liquid crystals, paving the way for the computational design of next-generation energy and smart material devices.