How do Toads in Naruto produce Oil?
The Toad Oil Bullet Is a skill used by Jiriya that converted his chakra into Toad oil. He spit out a volume of oil more than his body's size, proving that he could not have stored it inside his body. So the answer appears to be that there is a technique that converts chakra to Toad oil.

However, It is entirely possible the Toads, especially Larger ones, can swallow toad oil and regurgitate it later to conserve chakra usage required to create it. 

It is notable that the Toad oil they spit out and Toad oil used in Senjutsu training might be different oils. The wiki entry on Mount Myoboku notes:


  There is a sacred fountain on Mount Myōboku which produces a special oil which allows people to more easily feel the natural energy around them


So it seems only the fountan produces the Sage Mode oil. Also, the Sage Mode oil looks significantly like cooking oil, very light, clear, and of the same consistency, while The oil spit out to be ignited looks dirty, and dark, sometimes even opaque.

Edit: addressing your edit, There is only one thing that could possibly be added, and its a Theory.

You copied one line from the wiki on the trivia section of the Nature Transformation page. However, the very next line gives off a HUGE hint.


  Certain other techniques involved in manipulating substances are also often confused to be chakra natures, such as silk, bone, and ink. However, these substances cannot be produced from chakra alone, meaning the chakra is not changed in nature, and it is rather used to change already-existing substances.


The oil is called Toad Oil, So its could be some kind of substance a Toad can naturally create. If that is the case, then this would literally be exactly like Kimimarus Bone based Jutsu, Shikotsumaku, or Kidomaru's Spider Technqiues. Just like how They can take the natural substance that already exists (bone/Spider Silk) and use Chakra to create more of it, Toads would do the same. Perhaps that is what Kneading into a substance means, Mixing Chakra with a substance, and converting the chakra into more of that substance. Since everyone who uses Toad Oil is either a Toad, or Jiriya after he goes Sage mode and becames part toad, then they would likely have a natural source of it to duplicate via Chakra kneading, if that is indeed the case.