We report the synthesis, crystal structure, vibrational and luminescence properties of two heterometallic perovskite-type metal–organic frameworks (MOFs) containing the ammonium cation (NH4+, Am+): [NH4][Na0.5Cr0.5(HCOO)3] (AmNaCr) and [NH4][Na0.5Al0.475Cr0.025(HCOO)3] (AmNaAlCr) in comparison to the previously reported [NH4][Na0.5Al0.5(HCOO)3] (AmNaAl). The room-temperature crystal structure of AmNaCr and AmNaAlCr was determined to be R. The hydrogen bonding (HB) energy calculated using density functional theory (DFT) agrees well with experimental data, and confirms the existence of almost identical H-bonding in AmNaCr and AmNaAl, with three short hydrogen bonds and a longer trifurcated H-bond. Temperature-dependent Raman measurements supported by differential scanning calorimetry show that AmNaCr does not undergo any structural phase transitions in the 80–400 K temperature range. The high-pressure Raman spectra of AmNaCr show the onset of two structural instabilities near 0.5 and 1.5 GPa. The first instability involves weak distortion of the framework, while the second leads to irreversible amorphization of the sample. High-pressure DFT simulations show that the unit cell of the AmNaCr compound contracts along the c axis, which leads to a shortening of the trifurcated H-bond. The optical properties show that both studied crystals exhibit Cr3+-based emission characteristic of intermediate ligand field strength.