Do note that another term for absolute anonymous donations is "funding terrorism". So good luck hiding your donations entirely.
I suspect a sibling comment is correct - buy a visa/etc giftcard (perhaps with cash) and use that to donate online if the recipient accepts online payments - or send the card details via email, or the card via snail mail?
Depending on the recipient's fraud checks you can simply use your credit card but use a fake name & address; if they don't decline the transaction then that's all you have to do.
I suspect a sibling comment is correct - buy a visa/etc giftcard (perhaps with cash) and use that to donate online if the recipient accepts online payments - or send the card details via email, or the card via snail mail?