Calculating observable properties of chemical systems is often classically intractable and widely viewed as a promising application of quantum information processing. Here, we introduce a new framework for solving generic quantum chemical dynamics problems using quantum logic. We experimentally demonstrate a proof-of-principle instance of our method using the QSCOUT ion-trap quantum computer, where we experimentally drive the ion-trap system to emulate the quantum wavepacket dynamics corresponding to the shared-proton within an anharmonic hydrogen bonded system. Following the experimental creation and propagation of the shared-proton wavepacket on the ion-trap, we extract measurement observables such as its time-dependent spatial projection and its characteristic vibrational frequencies to spectroscopic accuracy (3.3 cm-1 wavenumbers, corresponding to >99.9% fidelity). Our approach introduces a new paradigm for studying the chemical dynamics and vibrational spectra of molecules and opens the possibility to describe the behavior of complex molecular processes with unprecedented accuracy.