I don't see how this could prevent it from starting, but I think the relay is for the HEATED oxygen sensors. O2 sensors have to be pretty hot to work properly--700 degrees or so I think. The heaters built in raise the temperature much more quickly than just heating in the exhaust manifold would do. Have you tried replacing the relay or substituting another relay, such as for the A/C or something (if it fits), to see if this allows it to start?
I think when you bypass the relay, that you may be backfeeding another circuit that has elements that are common to the circuit that you're bypassing, and that this circuit is critical to the engine starting--eg. fuel pump, ECU, etc. If all the fuses are good, both under the hood and under the dash, then I think you need to find a wiring diagram for the van and see what else is connected to this.