Complex optical systems such as high-quality microcavities enabled by advanced lithography and processing techniques have paved the way to various lightmatter interaction (LMI) studies. Sub-micrometer-precise lithographic development of a polymer photoresist allows construction of microcavity structures for various spectral regions based on the material's transparency and the geometrical sizes. On the other hand, this approach also avoids lattice-matching constraints in epitaxy, complex coating techniques, and shaky open-cavity constructions. Herein, a new approach based on 3D nanowriting in a photoresist is introduced, which can be used to achieve microscopic photonic Fabry-Pérot cavity structures with mechanically tunable resonator modes and polymer/ air-Bragg mirrors, directly on a chip or device substrate. By transfer-matrix calculations and computer-assisted modeling, it is demonstrated 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.