Quantum Computing enhanced Electron Microscopy (QCEM) aims to supplement the classical detector in electron microscopy (EM) with a quantum computer consisting of a trapped-ion lattice. This approach opens the door to a world of novel sensing and imaging techniques. The key ingredient to their development is a thorough understanding of the coupling of free electrons to the vibrational modes of a trapped-ion quantum computer. Here we present initial results, evaluating the scattering matrix elements in different coupling strength regimes, demonstrating the applicability of approximations, and exploring possible use cases of the coupling of a free-electron beam to a quantum computer.