I’m not sure where you are supposed to lubricate it. I would just try to work a little dab of lubricating grease into every moving part that you can reach with the aid of a tooth pick. You might need to disassemble the handle at the top pivot.

Petro Gel and Lubri-Film are two NSF-approved food-safe lubricants that I use on mixers, grinders, keg parts, and so forth. You don’t really need a food-safe lubricant for your corker though. A common lithium or molybdenum heavy grease should do fine. The only parts of the corker that come in contact with your bottles or corks are the plunger and the clamping jaws, and neither of those need to be lubricated.