Step 1: Identify the relevant constitutional provision.
Article 56 of the Indian Constitution deals with the term of office of the President; Article 56(1)(a) specifically lays down the procedure when the President wishes to vacate office by submitting a resignation.
Step 2: State the constitutional rule.
The President addresses the resignation letter to the Vice-President; the Vice-President is the constitutionally designated successor and holds the office of Acting President during any vacancy in the President's office.
Step 3: Eliminate other options.
The Chief Justice, Speaker, and Prime Minister are not the correct recipients under the Constitution; Article 56(1)(a) clearly mandates that the Vice-President alone receives the President's resignation.
\[ \boxed{\text{Vice-President}} \]