Starting from chiral nuclear interactions, we evaluate the contribution of the leading-order contact transition operator to the nuclear matrix element (NME) of neutrinoless double-beta decay, assuming a light Majorana neutrino-exchange mechanism. The corresponding low-energy constant (LEC) is determined by fitting the transition amplitude of the $nn\to ppe^-e^-$ process to a recently proposed synthetic datum. We examine the dependence of the amplitude on similarity renormalization group (SRG) scale and chiral expansion order of the nuclear interaction, finding that both dependences can be compensated to a large extent by readjusting the LEC. We evaluate the contribution of both the leading-order contact operator and standard long-range operator to the neutrinoless double-beta decays in the light nuclei $^{6,8}$He and the candidate nucleus $^{48}$Ca. Our results provide the first clear demonstration that the contact term enhances the NME by 43(7)% in $^{48}$Ca, where the uncertainty is propagated from the synthetic datum.