Chainsaw oil has got the sticky stuff in it, and logically, would not have detergents in it to work with filters in circulation systems like motor oils. In other words, it's pretty close to way oil. I never thought chain oil would ever be regarded as the "expensive" option. Some folk have used chain oil in place of way oil, though I am sure there will be opinion here on why that is not really a great thing to be doing.
When we say "vegetable oil", we mean, I guess, like cooking oil, or derived from cooking oil. If it lubricates, and does not have to handle extreme pressure (like in a differential axle), nor temperatures high enough to make it cook into carbon, then why not?
Originally, before we started cracking petroleum mineral oil into alkylate gasoline, we had "vegetable oils", and "mineral oils". Maybe we still do? My Dad used to buy a brand called "William Penn", and also "Duck" brand oil, and he told me they were oils of a vegetable nature. There is also, of course, the famous "Castrol R", as used for racing, and in aero engines. It came from caster oil, and it had that familiar aroma. I always thought the smell was much more pleasant than the stink of regular BP oil. My Dad told me that regardless the nicer smell, the rings on my car might be kaput, or maybe the valve guides.
Sure you can use vegetable oils to lube a chainsaw, but you might want to discover a substance to give it that "tacky" quality.