The ionization of molecular systems is important in many chemical processes such as electron transfer and photoelectron spectroscopy. Strong coupling between molecules and quantized fields (e.g. inside optical cavities) represents a new promising way to modify molecular properties in a non-invasive way. Recently, strong light-matter coupling has shown the potential to significantly improve the rates of hot electron driven processes, for instance water splitting. In this paper, we demonstrate that inside an optical cavity the residual interaction between an outgoing free electron and the vacuum field is significant. We further show that, since the quantized field is also interacting with the ionized molecule, the free electron and the molecular system are correlated. We develop a theoretical framework to account for the field induced correlation and show that the free electron-field interaction has sizeable effects on the ionization potential of typical organic molecules.