Hi Christina! Don’t you love your Pilot? I love mine! I use sewing machine oil. It’s not “eco-friendly” but it is light and doesn’t get too stinky or gooey. I use kitchen vegetable oil for cleaning up ink disk and rollers (plus a final wipe with odorless mineral spirits), but I don’t think vegetable oil would hold up as a lubricant on cast iron and steel. Would be interested to know what you find!
When I was a little younger we could buy engine oil that was recycled. It was just regular oil from our car engine that had been strained of all the gunk. The oil doesn’t actually wear out, it just gets very dirty and full of nasties. Maybe you can still buy it. Or even strain your own old engine oil, through fine material or something. I’m sure google will tell you how.