You may be able to do it but it would take so many CO2 cartridges that it wouldn't make sense financially. It would also be very difficult to get your carbonation dialed in without accurate psi control.
It would be much cheaper to get a 5lb CO2 bottle and regulator. I think it takes like 2-3 cartridges just to push a keg of carbed beer. I don't want to know how many it would take to force carbonate it.
Another option is to naturally carb in the keg with priming sugar and push the beer with the charger. Although, in the long run, I think you'll be happier with a CO2 tank and regulator.