Why won't my oven open its gas valve?
1) Pilot light is on


If the pilot light ignites and stays lit, then it cannot be a gas supply issue, the same supply line would feed the pilot light and the main burner, it has to. But the pilot light has a device called a "thermocouple"  which has a sensor that sits IN the flame and generates a small voltage when it is heated up that serves two functions;
A) It signals the pilot light gas valve to turn off if the pilot light is NOT keeping it hot (meaning it was snuffed out by a guts of wind or spilled liquid or something), and 
B) It signals the oven burner gas supply valve that the pilot light is lit, ready to ignite the burner.

If the thermocouple is old / corroded / malfunctioning, but is still giving ENOUGH output for the PILOT valve, but not enough for the MAIN valve, or the main valve is malfunctioning and/or not getting the voltage from the thermocouple, then you will get the exact symptoms you are describing. Your knob is signalling the main valve to turn on, which is the clicking you hear, but the main valve is not responding because it is not seeing the signal telling it that the pilot light is there to igniote it, so as a safety measure, it will not allow gas to go through.

As mentioned earlier, DIY novices and gas ovens are not compatible because the result of mistakes can be deadly. I would keep trying to find a repairman that will work on your stove. But keep in mind, they may ALREADY know that it's a bad valve and they know there are no replacements available for old Okeefe an Merrit ovens.