below is the wireing pics on how the back up lamps work ..
as you can see the back up lamps are powered up by fuse 12 in the interior fuse panel ,so lets start there with your volt meter ,, and with the key ON this fuse should have power to it on both ends of the fuse ,
if this fuse is ok ,, then keep the key ON ,, and get under the van , by the drivers door and look at the transmission ,, on there is the DTR,, unplug this switch and see if the Violet / orange wire has power on it comeing from the fuse 12.
if you find that you have power on this wire comeing from the fuse ,, then lets use a paper clip or a jumper wire and jump the violet / orange wire in the conector to the black / pink wire ,,,, this will by pass the switch and the back up lamps should come on ,,, if they do ,, then you know the DTR switch is bad and its has to be replaced.
any more info on this , just ask , we can go back and forth on this page
1. ) The DTR can get filled with water and short out ,,
2. ) if you have any add on trailer wireing , i would check this for shorts..
3.) if this is a cargo van with no panels inside of it , look for something in the van by the tail lamps , shorting out the back up lamps