Should work... Make sure that if it's an autotransformer you use, that the common tapping is connected to the neutral side (which should be connected to car chassis by the inverter)
Otherwise instead of having 110 and 0 volts relative to car chassis, you will have 240 and 130 volts.
You will still get 110 volts to run the sharpener - it's just not as safe as you may think!
Oh and if it's an AC motor in the sharpener, the motor will run at 83% of normal speed, as the inverter will be 50Hz, the sharpener will be designed to run on 60Hz.
It's likely cheaper to get a 110V 60Hz inverter....good old ebay etc...
Simon.