"Where does your contribution go?" is correct.
"does" is the third-person singular present tense form of the verb "to do" (I do, you do, he/she/it does, we do, they do).
"To do" is an example of an auxiliary verb, which I prefer to call a "helping verb", because that's what it does -- it helps. Some verbs are always helping verbs (for example, can, must, may...), while other verbs can sometimes be helping verbs and sometimes main verbs (for example, to do, to have...). Most verbs are main verbs only (to play, to sing, to go...).
In a normal sentence, the usual construction is: Subject, helping verb, main verb, followed by everything else. Here are some sentences, with the helping verb underlined:
Sue is playing the piano.
Peter can swim very well.
They have eaten the cake.
But this sentence doesn't have a helping verb:
Your contribution goes to people in third-world countries.
When we want to form a question, we usually (but not always) put the helping verb before the subject, like this:
Is Sue playing the piano?
Can Peter swim?
If we have a "question word" (an "interrogative pronoun" to be exact), that usually goes at the beginning of the sentence, but we still have the word order helping verb - subject - main verb:
What have they eaten?
But if we don't have a helping verb, how can we form a question? As I said, a question is formed by swapping subject and helping verb, but look at these two sentences:
Peter watches TV.
* Peter watches TV?
(The * means "This sentence is incorrect").
In some languages, like Spanish, you can do that -- just use the normal word order and use a question mark when writing, and intonation when speaking. (Spanish also uses the character ¿ at the beginning of a question to help readers recognise that this is a question.) But that is not formal English -- at best, it's informal; at worst, it's slang. We must have a helping verb.
So, to the rescue comes: "to do". This verb acts as a helping verb when a helping verb is necessary but no other helping verb works. It's a "dummy verb"; it doesn't mean anything, it's just there for the grammar.
When used in this way, "to do" takes the basic form of the main verb: this may be called the "verb word" or the "infinitve without 'to'":
I go; I do go
you go; I do go
he/she/it goes; he/she/it does go
Look at it this way: because the "-s" is now attached to the helping verb, we don't need it on the main verb any more -- one "-s" is enough.
So "Peter watches" becomes "Peter does watch", and now we can change the word order to helping verb - subject - main verb: "Does Peter watch TV"? Or: "What does Peter watch?"
So your example:
Your contribution goes to... (Oops! No helping verb. So let's use "to do")
Your contribution does go to... (So now we can make a question)
Does your contribution go to...? (And now we add a question word)
Where does your contribution go?
In the same way, we need a helping verb when making negative sentences, because a negative is formed by adding "not" (often shortened to "n't") to the helping verb:
Sue isn't playing the piano.
Peter can't swim very well.
They haven't eaten the cake.
Peter doesn't watch TV.
So, although ordinary sentences can work without a helping verb, questions usually can't (there are some that do work without a helping verb, though) and negatives can never work without a helping verb. If no helping verb is available, then "to do", wearing its "Superverb" costume, appears out of the blue and saves the day.