Coconut Oil would be fine bird-wise, but it could also make your hair really greasy...I don't know where you live, but if you have access to a grocery store that has a "Natural Living" department or whatever you want to call it, that's where I would look...I live in PA so we have a Wegman's grocery store here (throughout the entire Northeast US, based out of upstate NY), and they have a HUGE section in all of their stores that has all departments squeezed into one little area, such as food, medical, vitamin/supplements, cleaning, and then a huge double-aisle that is all "Personal Hygiene" products, including soaps, lotions, shampoos and conditioners, hair-care products, deodorants, shaving products, makeup, etc. Everything sold in this section, from the food to the cleaners to the hair-care products is All-Natural and most of it is also Organic, and most of the products are made from plants, earth/clay, etc. And it's also all "unscented" except for any natural scents that come from the natural ingredients...I buy a lot of different soaps their, everything from anti-bacterial hand soap to bars of soap to bathe with, as well as lotions, and then pretty much all of my cleaners. And there are a ton of hair-care products, from shampoo and conditioner to gel, mousse, paste/wax, hair spray, etc. So I guarantee you could find something there to replace each of your regular hair-care products, no trouble at all. And everything that I've ever purchased works great...I'm sure that there are other grocery stores that have the same type of "All-Natural Living" sections, maybe WholeFoods or Harris Teeter, or similar stores...Wegman's is awesome for that kind of thing...
***The only thing that you need to watch-out for is that if you do find a place that sells all-natural hair-care products, soaps, lotions, etc. that whatever you buy does not have any AVOCADO OIL as an ingredient! It's used in a lot of natural hair-care products, lotions, soaps, and even in a lot of the natural, plant-based cleaning products...and as we all know Avocado is toxic to birds...