The term "Castile soap" is used in this community to refer to a lye soap with no other oil than Olive oil, which is unlikely to be the only thing you need, though I understand some companies refer to any lye soap as "Castile." For the remainder of my post, I will be referring to lye soap as soap.
So I’m here asking if anyone makes and would sell me a completely coconut free Castile soap?
Making a lot of soap for someone you haven't worked with before is a bit of a risk, so I'll spell out what you need and why so you can see if anyone is already making it.
1. Shampoo- Lye soap should almost never be used for hair because the pH levels will damage your hair leading to dullness, thinning, and breakage. There are few people who get away with it and so some people do formulate bars for it, but if it ends badly, you've been warned. There are a lot of mild synthetic detergent shampoos that don't use coconut-based cleansers, but it does take some searching.
2. Foaming hand soap- you would need a liquid hand soap formulated specifically for foaming dispensers.
3. Dishes, cleaning, laundry- you would be looking for soap that's almost entirely babassu, palm kernel oil, or another extremely cleansing oil, formulated specifically with that type of use in mind. This type of soap will almost certainly irritate your skin and most individuals who use this type of soap handle it with gloves on. You may need to use an acidic rinse on whatever you use it on to ensure all the soap is off and will not irritate your skin on contact.
4. I'm assuming "all the things" refers to showering, and for that you would need a mild soap formulated for use on the human body. This is pretty much the only item on your list for which a Castile soap as understood by this community could possibly be what you're looking for, but there are plenty of other mild soaps that don't use coconut to choose from as well.
Attempting to use soap formulated for the human body on inanimate objects will usually leave a residue from the aspects of the soap that are formulated to keep it from taking all your skin oils and leaving you with dry skin.
Anyways, you're at least looking for 3 different products if not 4 or more, so now you can tailor your search- and expectations- accordingly.