To is a preposition, which isn't generally supposed to go at the end of a sentence. I would use the first one.
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eagerness"This is the name of the person to whom this letter needs to be sent or,Both are right; the first is quite formal. If you don't like either sentence, you have two more options: This is the name of the person [who/that] this letter needs
"This is the mame of the person whom this letter needs to be sent to"