Molecular electronics is a rapidly developing field focused on using molecules as the structural basis for electronic components. It is common in such devices for the system of interest to couple simultaneously to multiple environments. Here, we consider a model comprising a double quantum dot (or molecule) coupled strongly to vibrations and weakly to two electronic leads held at arbitrary bias voltage. The strong vibrational coupling invalidates treating the bosonic and electronic environments simply as acting additively, as would be the case in the weak coupling regime or for flat leads at infinite bias. Instead, making use of the reaction coordinate framework, we incorporate the dominant vibrational coupling effects within an enlarged system Hamiltonian. This allows us to derive a nonadditive form for the lead couplings that accounts properly for the influence of strong and non-Markovian coupling between the double dot system and the vibrations. Applying counting statistics techniques, we track electron flow between the double dot and the electronic leads, revealing both strong-coupling and nonadditive effects in the electron current, noise, and Fano factor.