At Taylor Park, near Ridgeland and Division in Oak Park, there is an area with grills (actually on the Northern end of the park, along Berkshire). Essentially, its BYOC--bring your own charcoal. There are some picnic tables in the same vicinity.
Now, let me say, first, I have seen people using the grills, so I know its doable, but (but), I have no idea how it works. I do not know if it is first come first serve. I do not know if you need a reservation or permit.
But the whole thing DOES seem possible.
Rob
Think Yiddish, Dress British - Advice of Evil Ronnie to me.