Right idea, just need a little more.
It is totally legal to retrofit a ground wire anywhere you wish. I would (de-energize everything and) run the ground wire starting at the service panel (or appropriate point). I would pause when I was near the offending box, and light everything back up. Then, having a voltmeter, I would set it on AC volts and measure between the almost-there ground wire and the offending junction box. A voltage reading indicates a ground fault in a circuit using the box**.
** Keep in mind that wires can be like antennas, and a very sensitive voltmeter (like most cheapie DVMs today) can detect what is essentially a radio signal being picked up from parallel wires. This has no useful force to it, but will produce a reading as high as 90% of line voltage. A low-impedance measuring device, such as one with a mechanical gauge or neon light, will not be fooled.
That ground fault is serious business and needs to be fixed. It could be in the wiring or in a grounded appliance.
And then ground the box
Once you've done the test, I would continue to use the test to chase down the ground fault.
Tell everyone with computers to save their work, then shut off half the breakers, go back and measure if the ground fault is present, shut off the other half. Once you determined the half of breakers which must be on to cause it, turn half of them off, continue to halve your breakers until you find the one breaker that, turned on or off, causes the ground fault to occur or not. Now you're down to one circuit and work it from there.
That metal box is your friend if it's grounded. The idea of replacing it with a plastic one is "hiding the symptom", you still have a dangerous ground fault. Plastic boxes are not done to insulate, they are done because they are super cheap. Like 30 cents instead of $1.10. The building industry convinced the NFPA that they are not that much less saf in residential. In commercial work you must use metal.
If your house is not grounded generally, it is absolutely fine and a rather good idea to retrofit grounds throughout the house, running a spiderweb of ground wires to each junction box. Grounds served out of the same service panel are allowed to share the same wire. So you only need to run a heavy 10 AWG backbone to the stove/water heater/dryer sockets, then you can tap off the backbone for all your smaller 12 and 14 aWG grounds. These splices need to be made inside electrical boxes, so add a few in unfinished areas.
You can also do most of the wire-mongering with the power on, and only turn off power for the final connection to the junction box (in case of this very type of situation).
This, with AFCI breakers, and GFCI receptacles where appropriate, will make your house absolutely state-of-the-art in terms of electrical safety.