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The last decades witnessed the emergence of the field of correlated disordered media, a great challenge offering a large panel of new perspectives for applications in theoretical modelling and material fabrication. The efficient design of structures with a controlled level of spatial correlation is a central challenge in this field, in a context where existing techniques generally rely on gradient descent on non-convex functions and on the use of stochastic methods to explore vast design spaces more efficiently. In this work, we propose a new generative technique based on Alan Turing's morphogenesis theory for designing correlated disordered materials. Inspired by the structuring of living organisms, this technique relies on the definition of simple local interactions guiding the self-organization of a generated medium. The decentralization of design constraints and the elimination of cost function minimization make this approach natively scalable to the design of large domains with controlled levels of disorder. As a validation, the morphogenetic generation of stealthy hyperuniform disordered structures is exploited to reproduce an experiment of isotropic electromagnetic bandgap synthesis in the microwave range using a low refractive index contrast.