We identify the existence of nonlinear optical (NLO) activity in a number of novel $ABX_3$-type metal-free perovskites, where $A$ is a highly tuneable organic cation, $B$ is a NH$_4$ cation and $X$ a halide anion. Through systematic first-principles calculations, we identify important trends to chart the second-harmonic generation of this class of materials. We study three perovskites MDABCO-NH$_4$I$_3$, CNDABCO-NH$_4$I$_3$ and ODABCO-NH$_4$I$_3$ for use as deep-UV second-harmonic generation materials. We identify the role of the dipole moment imparted by the organic group on the $A$ cation as an important parameter to tune the NLO properties of these materials. We apply this knowledge functionalising the organic group DABCO with the highly polar cyanide CN$^-$ group, and we demonstrate a significant improvement of the NLO response in this family of materials. These findings can accelerate the application of metal free perovskites as inexpensive, non-toxic, earth-abundant materials for the next generation of optical communication applications.