WOW! That's an OLD slicer made back in the 1930's to 50's! Cool!
YES. For safety reasons that weren't observed back then, securely connect the green (or green/yellow) wire to the slicer's chassis. As such, should the hot (black) wire inadvertently short to the frame by way of a rub-through, it will trip a circuit breaker (which you WANT it to) instead of making the frame a 115v shock hazard.
Be sure that the hot and neutral wires are routed appropriately so they don't get rubbed through due to vibration or from contact with moving parts. Along with suitable routing to avoid contact with moving parts, you can use wire ties and/or automotive plastic conduit (loom) to protect wiring you feel might be susceptible to rub-though from contact with sharp edges within the chassis.
I ALSO suggest to NOT use wire nuts when installing your new power cord since those can work loose from vibration. There are several styles of crimp-type butt connectors you can use. I like the shrink-tubing type, but conventional butt connectors can be used too when filling in the connector ends with some dielectric grease to keep moisture out. Those can ALSO be gotten from an auto parts store.