Free electrons serve as quantum probes that can become coherently correlated with other quantum systems, offering access to advanced metrological resources. We present a scheme that allows electrons to be coherently coupled to a trapped-ion quantum computer via the Coulomb interaction. By preparing the trapped ions in non-classical states of motion, an effective Pauli-X interaction between the free electrons in a superposition of paths and the qubits can be realized [1]. This naturally leads to entanglement between the quantum computer and the state of the electron, which may be used to transfer information about a specimen to the quantum computer, allowing for coherent information processing and readout using arbitrary measurement bases. Correlated measurements on the trapped ions and the electron give access to interesting probe states with orbital angular momentum and discrete symmetries. Multiple electrons can become entangled through their interaction with the quantum computer, enabling sensing schemes that go beyond the standard quantum limit and may pave the way to dose-efficient electron microscopy.