Sounds like you might want to look into Donor Advised Funds
At MIRI we receive donations through these types of orgs, Fidelity Charitable being the most common.
Are you worried about snail mail spam or just email spam? The latter is easy to deal with. I have a dedicated email address which is the one I give out to computers that ask for one, to take the spam away from my others. Also, what charities are you afraid of spam from? In the year I've been donating to MIRI, I've gotten only one random email from them. Givewell also has this, which will pass on your money in their recommended split among their top charities. Seeing as they seem to be all about donating to a charity because it is the best and you want to do good, and not because you have received guilt-inducing spam directing your donations to a particular charity, I would be surprised if they spammed their donors. Perhaps someone who donates to them can confirm.
The most pernicious charity spam I've received lately is from the bone marrow donor registry I signed up for about 18 months ago. I'm not sure that would work in this circumstance.
There are some underlying differences between most charities and bone marrow donation, though. The registry gets no utility from prospective donors until they match donors with patients, so if they can't reach a prospective donor, that database entry is worthless (except perhaps to artificially inflate the size of the database and attract financial donations). A charity that's already gotten some of your money has already gotten some utility, and would like to have more, but faces less risk from losing contact with people on their list.
Also, can you even sign up for bone marrow donation anonymously? I suppose you could give them fake information but they'd still need to be able to reach you.
I can't think of very many other charities that exist to match non-fungible donations with people who need them, though -- pints of O+ are interchangeable, as are hours of unskilled labor -- so I think you've found a really interesting outlier.
One thing that sometimes makes me hesitate to donate to a cause is that, unless you're donating in person and using cash, you're inevitably signing up for a gigantic stream of junk mail, not just from the organization you gave money to, but other, often totally unrelated charities as well. I haven't noticed a lot of these charities offering a privacy policy that lets you avoid this, but I haven't paid close attention because frankly, I don't think I'd have a lot of confidence in such a privacy policy even if I saw one in some literature.
I wonder if there are donations to be gained in guaranteeing this sort of privacy by going through a third party. Charities could include the usual pre-addressed envelope in their mailings, only instead of their own address it would go to an organization called Givepal. The envelope would include the charity's id, and donors would be instructed to make their checks out to Givepal, who would then distribute the money to the specified charity, keeping the transaction anonymous. Givepal could survive by taking a cut of the donations if necessary, or could itself operate as a non-profit.