Complex optical systems such as high-quality microcavities enabled by advanced lithography and processing techniques paved the way to various light-matter interactions (LMI) studies. Without lattice-matching constraints in epitaxy, coating techniques or shaky open cavity constructions, sub-micrometer-precise lithographic development of a polymer photoresist paves the way to polymer microcavity structures for various spectral regions based on the materials transparency and the geometrical sizes. We introduce a new approach based on 3D nanowriting in photoresist, which can be employed to achieve microscopic photonic Fabry-Perot cavity structures with mechanically-tunable resonator modes and polymer/air Bragg mirrors, directly on a chip or device substrate. We demonstrate by transfer-matrix calculations and computer-assisted modelling that open microcavities with up to two air-Bragg reflectors comprising alternating polymer/air mirror-pair layers enable compression-induced mode tuning that can benefit many LMI experiments, such as with 2D materials, nanoparticles and molecules.