Eldon
You asked the question if neem oil is organic and is it safe for hummingbirds. I found this info and if you go to this link and scroll down to where it says "safe insecticide-Neem oil" I think it will ease your mind.
Personally I have no problem using it nor fear any harm may come to the birds and I spray the stuff all over the leaves and directly into all blooms and flowers because that is where the aphids hide.
http://www.infobarrel.com/Safe_Insecticide_-_Neem_OilSteve
Martinsville, In
Heat zone 6
Sunset zone 35