Never did this before, but it seems all wallet services want your bank information at some point.
This is pure nonsense. The only ones that eventually may or may not require your information are
online wallets. They are just a sub-category of a lot of wallets (mobile, desktop, paper, hardware, etc.).
Are there any wallets that don't require it?
Any desktop wallet, e.g. Electrum.[1]
As in, can I fund them with a Vanilla Visa card (just an example) and then later when I want the money in my hands, can I withdraw to a bitcoin ATM, also without showing ID or giving my bank account info?
I have no idea what you're trying to say.
[1] - It is generally recommended that you keep your money on your own machines. If you are purchasing online, say via an exchange, you should withdraw as soon as possible.