Could be the ignition solenoid. Easy test: When this happens, does the dash HVAC fan (assume it only runs when ignition on) run?
If not, locate the ignition solenoid (probably up in the dash area or perhaps in the forward most basement compartment in front of the driver's wheel on some coaches).
Use your digital voltmeter to check voltage on each of the two large lugs. With the key on, both large lugs should read the same as battery voltage and 12+ VDC.
This is a fairly high failure rate item. And as they age, the contacts can fail-- work sometimes, not others.
If this is it and you need to move it, let us know. There is a "work around".