Utilizing a biomimetic feedback network, we design microcapsules that self-organize and undergo directed, self-sustained motion over a substrate. In our model system, microcapsules act as localized sources of chemicals that diffuse through the surrounding fluid medium. We specifically consider three types of microcapsules, each producing a different chemical species with production rates modulated by a regulatory network known as the repressilator: each species represses the production of the next in a cycle. This network has previously been studied in the context of gene expression regulation where all species are produced in a small, well-mixed volume of space. In this case, the dynamics can be described using ordinary differential equations and it is known that the levels of each species can approach constant values or exhibit large amplitude oscillations, depending on model parameters. In the current work, we present an analysis of the repressilator system with finite spatial separation between sources of each chemical component. Conditions are given for the steady and oscillatory regimes. We then extend the model to allow movement of the microcapsules over a planar substrate. The microcapsules are placed on a surface and the chemicals released into the fluid are adsorbed onto the surface, altering adhesive interactions between the surface and the microcapsules. Gradients in the surface energy resulting from this adsorption generate lateral forces, leading to self-induced motion of the microcapsules. We numerically simulate this system by combining the lattice Boltzmann method to solve the Navier-Stokes equation for hydrodynamics, the immersed boundary method to couple the microcapsule motion to the fluid flow, and finite difference methods for the advection-diffusion of chemical species. This model exhibits a diverse range of behavior, including several distinct modes of sustained motion of the microcapsules. We highlight the role of the repressilator regulation network and demonstrate how combining chemical sensing with motile response can lead to new behavior in the form of spatial organization.