Both your sentences are okay. It depends what you mean. Whom did he write this letter to? This is asking what person did he write and send a letter to. He wrote a letter to be received by XY. Whom did he write this letter for? This is asking what person did he write a letter for, because that person cannot write. He wrote a letter for XY because XY cannot write. Maybe XY has a broken arm, so someone else ("he") had to physically write the letter for XY. Note that the difference between "who" and "whom" is going out of style, especially in spoken speech. That is, in spoken speech, one is more likely to hear: Who did he write the letter to? or Who did he wrote the letter for? Additionally some teachers may follow an obscure rule that says that one may not end a sentence with a preposition. But this rule is ridiculous.

Whom did she write this letter to? To whom did she write this letter? This letter is for XY.

в каком предложении можно и так и так