Where can I find the letters of 24 December 1991 regarding the continuation of the membership of the USSR by the Russian Federation?

The UN Archives has provided a scan from the archives of the Secretary-General Javier Perez de Cuellar, folder “Union of Soviet Socialist Republics (USSR)", reference number S-1024-0174-0001. The archival material is addressed to the executive heads of the UN System. Please see the attached file for a scan of the letters in English and Russian.

This material was not issued as a UN document. 

In its section on membership, the 1989-1992 Repertoire of the Practice of the Security Council states, 

With respect to the Union of Soviet Socialist Republics, the Council recommended the admission, as sovereign States, of 12 of the 15 former constituent republics. Two were already members of the Organization. [footnote 5] The Russian Federation did not apply for membership. By a letter dated 24 December 1991, [footnote 6] the representative of the Union of Soviet Socialist Republics transmitted to the Secretary-General a letter of the same date by which the President of the Russian Federation informed the Secretary-General that the membership of the Union of Soviet Socialist Republics in the United Nations, including the Security Council and all other organs and organizations of the United Nations system, was being continued by the Russian Federation, with the support of the countries of the Commonwealth of Independent States. The President of the Russian Federation requested that the name “Russian Federation” be used in the United Nations, in place of the name “Union of Soviet Socialist Republics”. He affirmed that the Russian Federation maintained full responsibility for all the rights and obligations of the Union of Soviet Socialist Republics under the Charter of the United Nations, including the financial obligations. The Secretary-General informed the President of the General Assembly and the President of the Security Council of these letters and steps were taken to inform all organs and organizations of the United Nations system. The position of the Russian Federation was not challenged.

[footnote 5] Belarus and Ukraine.
[footnote 6] Not issued as a United Nations document. For a summary of the letter, see Official Records of the General Assembly, Forty-seventh Session, Supplement No. 2 (A/47/2), p. 277.

(Source: Repertoire of the Practice of the Security Council, Supplement 1989-1992, page 228, emphasis added (pdf))

A/47/2 is the Report of the Security Council to the General Assembly, covering the period from 16 June 1991 to 15 June 1992. Page 277 contains an endnote to Part IV of the Report, "Matters brought to the attention of the Security Council but not discussed in the Council during the period covered":

Notes
1/ By a letter dated 24 December 1991, the Secretary-General requested the President of the Security Council to bring to the attention of the members of the Council the text of a letter of the same date from the Permanent Representative of the Union of Soviet Socialist Republics, transmitting to the Secretary-General a letter, also of the same date, from Mr. Boris Yeltsin, President of the Russian Federation, in which he informed the Secretary-General that the membership of the Union of Soviet Socialist Republics in the United Nations, including the Security Council and all other organs and organizations of the United Nations system, was being continued by the Russian Federation with the support of the countries of the Commonwealth of Independent States. He requested that the name "Russian Federation" should be used in the United Nations in place of the name "Union of Soviet Socialist Republics" and stated that the Russian Federation maintained full responsibility for all the rights and obligations of the Union of Soviet Socialist Republics under the Charter of the United Nations, including the financial obligations. He also requested the Secretary-General to consider that letter as confirmation of the credentials to represent the Russian Federation in the United Nations organs for all the persons currently holding the credentials of representatives of the Union of Soviet Socialist Republics.