Badge machine designs vary, but essentially you need to lubricate all the "metal on metal" parts that slide or move against each other.

Example; where the handle pivots on the main body.

What lubricant to use? we recommend a 'standard' oil, something like sewing machine oil, it's mineral based and not too heavy.
Do not use a penertrating type fluid/lubricant like WD40, that product is good for many uses including the reason it was named "Water Displacement, 40th formula", but it tends to be too light and you need something that stays lubricating  for longer.

I would normally suggest a dab of oil about every 500 badges or once every 6 months if your machine is only used occasionally.