Vertical stacking of different two-dimensional (2D) materials into van der Waals heterostructures exploits the properties of individual materials as well as their interlayer coupling, thereby exhibiting unique electrical and optical properties. Here, we study and investigate a system consisting entirely of different 2D materials for the implementation of electronic devices that are based on quantum mechanical band-to-band tunneling transport such as tunnel diodes and tunnel field-effect transistors. We fabricated and characterized van der Waals heterojunctions based on semiconducting layers of WSe₂ and MoS₂ by employing different gate configurations to analyze the transport properties of the junction. We found that the device dielectric environment is crucial for achieving tunneling transport across the heterojunction by replacing thick oxide dielectrics with thin layers of hexagonal-boronnitride. With the help of additional top gates implemented in different regions of our heterojunction device, it was seen that the tunneling properties as well as the Schottky barriers at the contact interfaces could be tuned efficiently by using layers of graphene as an intermediate contact material.