To this native US English speaker, either one is grammatically correct, but the second one, with wrote, is much simpler and more natural.
The writing of the letter is entirely in the past; it is a completed action. Therefore, the simple past ("wrote") is more appropriate. The present perfect ("has written") is (as the name implies!) actually a present tense which describes the fact that something happened in the past but continues (or could continue) until the present. But the writing of the letter is a discrete action that was completed entirely in the past, so it's simpler and more appropriate to say the person wrote the letter and he is the person who wrote the letter.
We often see English language learners overuse the perfect; a good general rule of thumb is something we here like to call FumbleFingers' Perfect Truism , which is "Don't use the perfect unless you really have to."