For milled machine parts, why not WD40? It's been used to refurbish antique motors and engines for decades. You can spray some at the end of a cotton swab or use the straw nozzle to get right between the joints. The only risk I can see is using too much and having it drip down onto the punch-arm and then the fabric.
WD-40 is not a true lubricant, it’s a penetrating oil that dries out pretty quickly and if used as the only lubricant it will likely gum things up as it dries. OP wants a light machine oil, something like 3-in-1 (which you can find a big box home improvement stores) or most sewing stores will carry sewing machine oils. Personally I’d go with something like Starret 1620 or similar. Anything with a drip tip is likely going to be easier to apply in controlled amounts as well.
WD-40 is not a true lubricant, it's more of a cleaner. It's great for penetrating stuck bolts or scrubbing rust but not for oiling moving parts. It dries out quickly and does little to protect wearing parts. For a sewing machine "light machine oil" or even just purpose made sewing machine oil is the correct option. When it's used to "refurbish engines" they're removing dirt and rust, while likely also polishing pitted areas. They will clean off the WD-40 afterwords and use motor oil the rest of the time.