We present a new technique for the upcoming tri-static incoherent scatter radar system EISCAT 3D (E3D) to perform a volumetric reconstruction of the 3D ionospheric electric current density vector field, focusing on the feasibility of the E3D system. The input to our volumetric reconstruction technique are estimates of the 3D current density perpendicular to the main magnetic field, $\mathbf{j}_\perp$, and its co-variance, to be obtained from E3D observations based on two main assumptions: 1) Ions fully magnetised above the $E$ region, set to 200 km here. 2) Electrons fully magnetised above the base of our domain, set to 90 km. In this way, $\mathbf{j}_\perp$ estimates are obtained without assumptions about the neutral wind field, allowing it to be subsequently determined. The volumetric reconstruction of the full 3D current density is implemented as vertically coupled horizontal layers represented by Spherical Elementary Current Systems with a built-in current continuity constraint. We demonstrate that our technique is able to retrieve the three dimensional nature of the currents in our idealised setup, taken from a simulation of an active auroral ionosphere using the Geospace Environment Model of Ion-Neutral Interactions (GEMINI). The vertical current is typically less constrained than the horizontal, but we outline strategies for improvement by utilising additional data sources in the inversion. The ability to reconstruct the neutral wind field perpendicular to the magnetic field in the $E$ region is demonstrated to mostly be within $\pm 50$ m/s in a limited region above the radar system in our setup.