- Vegetable oil as used in medieval lamps simply cannot burn in the open because the flashpoint is far too high (link).
- Use of it as a grenade overshadows existing low-level player powers (attacks and spells).
- It's unfair to have the power "hidden" in the DM's book where new players won't know about it, and
- It contradicts Gygax's thesis in the DMG that chemicals are less volatile in the fantasy world than real life (p. 32, 113).
Generally any combination of realism plus game design running in the same direction locks in a decision for my games (see golden rule: link), but here I've got a quadrafecta, so you might say that the proper ruling is massively overdetermined. You can see prior blogs for more of my gaming history and analysis of the subject (link), and also documentation of me actually running the experiment in trying, and failing, to get a pool of vegetable oil to burn (link). (Side note: This is not to say that some "Greek Fire" element in the campaign is totally out of place, but it must be intentionally designed and priced appropriately, not the same as mundane lamp oil, and not the result of a rules glitch or oversight.)
But one other possible consideration remains. Previously, poster DHBoggs was tremendously helpful in pointing out the following quote by Gygax in the ENWorld Q&A thread from 2007 (emphasis mine; currently archived here):
In OD&D the 1st level PCs did do several things to help extend their chances--hire men-at-arms, use missile weapons (including flaming lamp oil, that is kerosene), and run away when things appeared to be too dangerous to stay and fight.
While "kerosene" wasn't trademarked as such until 1854, and thus the name would be an anachronism in the middle-ages milieu, similar processes were described as far back as the 9th century (link). Thus, one might hold out hope that the basic D&D campaign has access to somewhat more advanced lamp oil technology, i.e., kerosene, which may be more successful in burning as a weapon in the open. So we should run that experiment and check -- although if you read the link above on "flashpoints", then you already know what the answer will be...
Today I've procured a quantity of kerosene and poured it in the bottom of this container.
Applying a lit match to the surface of the pool of kerosene does nothing (even on a fairly hot day); the flashpoint is too high for it too ignite (specifically, somewhere between 100 and 160 degrees Fahrenheit).
Even dropping the lit match directly in the pool of kerosene does nothing except burn the match itself.
Now, if we set up some element as a wick (which could be a bit of string or the match itself) and allow it to become saturated with the fluid, then it certainly does draw up the kerosene to maintain the flame, thus serving as a very nice source of light.
Actually, the wick burns like the dickens when I do this. I'd better put it out before I stink up my whole room. But usable as a weapon, grenade, or burning pool of oil? No, that's impossible.
In conclusion: There's definitely no substance that's both commonly used as lamp oil and has the capacity of igniting in an open pool (at normal temperatures). The very idea is just kind of ludicrous, akin to saying that one's pack-mule can naturally fly, or that one could live by eating nothing but belladonna (deadly nightshade). Maybe with some magical or exotic treatment it's an interesting element, but not out-of-the-box as a mundane, everyday item. Even if we posit some anachronistically more advanced substance like kerosene, as shown here, it still fails the test.