Ground Controller described operational deviation when clearing a vehicle to cross a closed runway; not realizing the vehicle location; the clearance resulting in an uncoordinated active runway crossing.

Narrative

Working ground control with Runway XXL closed along with numerous vehicles constantly calling for crossing taxiways to the closed runway. The LOA states that any vehicles wishing to cross an active runway must call the Local controller for permission. I was coordinating several different things at once for weather in the area and reroutes. The vehicle called me to enter the closed runway and stated his position; I misunderstood his call; I thought he was at the same location as the other 12 vehicles were that keep calling me. I told him to proceed on Runway XXL thinking he was on the other side at taxiway B1; the vehicle crossed Runway XXR and proceeded on Runway XXL. This airport has had a problem with vehicles for years due to construction and the facility letting the airport authority do as it pleased. There have been calls for getting vehicles off the ground frequency for years. The airport vehicles (OPS) have permission to roam the movement areas on their own; they should be the ones to escort any vehicles about the movement areas and all vehicles (except active emergency vehicles) should NEVER have to cross a runway since there are numerous perimeter roads that are usable for this.