I wonder if there is any way for any of the United Nations different parts to vote to remove the veto right from being used in the UN Security Council.
The right of the 5 permanent member of the Security Council to veto its decisions is enshrined in Article 27, paragraph 3 of the Charter of the United Nations:
Decisions of the Security Council on all other matters shall be made by an affirmative vote of nine members including the concurring votes of the permanent members
Because this is part of the Charter, it requires an amendment to change it. Article 108 specifies one method for doing this:
Amendments to the present Charter shall come into force for all Members of the United Nations when they have been adopted by a vote of two thirds of the members of the General Assembly and ratified in accordance with their respective constitutional processes by two thirds of the Members of the United Nations, including all the permanent members of the Security Council.
Article 109 specifies a different method, but with an identical restriction.
In other words, the veto can only be removed with the approval of two thirds of UN members, and the approval of all the veto holders. Hence a veto holder would have to consent to have its veto removed.
Even if the assembly could do this, the point raised by the US delegation when the current setup was being negotiated still stands, probably:
In the end, the great powers made it clear that their participation in the UN was contingent on them being accorded the veto over all but procedural matters. This point was emphatically made at San Francisco by US Senator Tom Connally, part of the US delegation. Connally famously lectured delegates from states questioning the veto, “You may go home from San Francisco…and report that you have defeated the veto…but you can also say, ‘We tore up the Charter!’” He then proceeded to tear up his copy of the draft Charter.
And I haven't delved into the legal details, but I bet the GA doesn't have power to regulate the Security Council, given its general lack of legal teeth:
The General Assembly votes on many resolutions brought forth by sponsoring states. These are generally statements symbolizing the sense of the international community about an array of world issues. Most General Assembly resolutions are not enforceable as a legal or practical matter, because the General Assembly lacks enforcement powers with respect to most issues. The General Assembly has authority to make final decisions in some areas such as the United Nations budget.
General Assembly Resolutions are generally non-binding on member states, but carry considerable political weight, and are legally binding towards the operations of the General Assembly. The General Assembly can also refer an issue to the Security Council to put in place a binding resolution.
Probably the closest case when GA tried to do something like you ask (take over some Security Council powers) is the "Uniting for Peace" GA resolution 377 A (V) of 1950. Which was invoked occasionally thereafter, but not so much recently:
A classic example of the majority cases in which Security Council referral took place is resolution 120 (1956) regarding the Soviet invasion of Hungary:
The Security Council,
Considering that a grave situation has been created by the use of Soviet military forces to suppress the efforts of the Hungarian people to reassert their rights,
Taking into account that because of the lack of unanimity among its permanent members the Security Council has been unable to exercise its primary responsibility for the maintenance of international peace and security,
Decides to call an emergency special session of the General Assembly, as provided in General Assembly resolution 377 A (V) of 3 November 1950, in order to make appropriate recommendations concerning the situation in Hungary.
From a historical perspective, there can be little doubt that the exercise of the veto by the Soviets on this occasion represented an abuse of the veto right, a fact that was clearly reflected in the Council’s adoption of the procedural resolution by 10 votes to 1 (the Soviets) in an 11-seat body, and the subsequent overwhelming vote of the General Assembly condemning the aggression. In light of that abuse, it could not be convincingly argued that the Security Council was upholding its responsibility for the maintenance of international peace and security. Even though the mere condemnation ultimately issued by the Assembly was apparently futile, and reasonably resulted from member states’ lack of political will to confront a superpower more robustly, the process gave a voice to the disenfranchised majority of UN members, and helped to retrieve some credibility for the system of collective security. [...]
the resolution’s initial lustre has dramatically faded from the perspective of the Western P5 members. This can partly be attributed to the fact that between 1945 and the conclusion of the first wave of decolonization in 1963, the General Assembly grew in size from a manageable body of 51, which could virtually guarantee a majority in favour of the West, to a massive body of 114. The vast majority of the new additions came from the developing world, and completely transformed the little Cold War Assembly of the 1950s into an independent force of its own, represented in large part by the so-called Group of 77 states, which today in fact consists of 131 of 193 total UN members. This expansion of the UN helped precipitate an abandonment of the resolution by the Western P5 members, which could no longer count on predictable majorities to carry their interests. Accordingly, UN practice has witnessed a shift back towards the absolute predominance of the Security Council, relegating the Assembly to relative impotence. In fact, following a flurry of four emergency special sessions convened at Security Council request between 1956 and 1960, the Council has only referred matters to the Assembly on two additional occasions, in 1980 (regarding Afghanistan) and 1982 (regarding Palestine). The resolution has virtually disappeared from the vocabulary of the Council for the past 30 years.
A somewhat similar situation occurred in 1974 when 3 permanent Security Council members vetoed the expulsion of South Africa, but South Africa was nonetheless suspended by the GA (91 to 22 votes) for the (then) current session, because that was within their powers.
Nov. 12—The General Assembly voted today to suspend South Africa's participation, in its current session.
The decision was without precedent in United Nations history, but it did not exclude the South African, Government from membership in the world organization itself. It means that the, delegation will not be pemitted to take its seats, speak, make proposals or vote. [...]
The United States, Britain and France used their veto power in the Security Council on Oct. 30 to block an African-led campaign to expel South Africa from the United Nations.