Electron tunneling provides the primary reaction channel for electron transfer (ET) in many molecular systems. The analysis of such systems therefore requires the consideration of electronic coherence and interference effects. A model system for which tunneling may be either symmetry forbidden or allowed is considered here in the presence of a driving infrared (IR) field. It was previously shown that inelastic tunneling allows ET in the symmetry forbidden system via vibronic interactions. We show here that explicit considerations of IR interactions with these systems further changes the ET kinetics, Analysis in the framework of Floquet theory reveals that interaction with an IR field may increase the probability of inelastic tunneling and thus enhance the ET rate for a system in which elastic ET is forbidden. It is shown that IR driving of a nuclear oscillator promotes the oscillator into excited states that couple more strongly to the tunneling electron. Furthermore, it is shown that IR driving may suppress the ET rate in this same system, depending on system energetics. In a model where elastic tunneling is symmetry allowed, we examine vibronic modulation of ET in the Floquet framework. ET rates are computed for symmetry allowed and forbidden model systems, and vibronic interactions are found to suppress or enhance ET in both systems. The inelastic ET rate may be enhanced over 4 orders of magnitude in the symmetry disfavored case or suppresed by 15% in the same system. Effects of IR on ET rates in the symmetry-allowed system are weaker with enhancements up to 34% over the undriven rate and suppression of about 3% with IR driving present. This study is the first theoretical and computational exploration of ET rate control by IR irradiation of the bridge.