One proposed reform of UNSC suggests that potential new candidates are Germany, Japan, India, and Brazil.
Does Saudi Arabia qualify for this position under any criterion?
One proposed reform of UNSC suggests that potential new candidates are Germany, Japan, India, and Brazil.
Does Saudi Arabia qualify for this position under any criterion?
There is no "application process" for membership in the UN security council. The permanent security council members are named explicitly in Article 23 of the UN charter:
The Republic of China, France, the Union of Soviet Socialist Republics, the United Kingdom of Great Britain and Northern Ireland, and the United States of America shall be permanent members of the Security Council.
The permanent members are permanent member because they are on the list in Article 23 (the Russian Federation has inherited the seat of the USSR because they are generally recognized as their political successor, and Republic of China is obviously a shorthand for the People's Republic of China, and whoever listens to those people on Taiwan who say otherwise gets into trouble with the one and only Chinese government which exists in the world).
There are no other official requirements or replacement processes. So adding or removing any council members would require an amendment to the UN charter, nothing more and nothing less.
The process of charter amendments is explained in Article 108:
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.
So if Saudi Arabia wants to propose an amendment to the UN charter which reads "The Kingdom of Saudi Arabia shall be a permanent member of the Security Council" they can try, but it might be difficult for them to get the complete current security council and two thirds of the whole UN to agree to that.
Instead of sponsoring their own amendment, they could also try to negotiate with the sponsors of other amendment proposals to get something in their proposal which might help them to achieve the goal of gaining more influence in the UN in the long term (for example, any amendment which suggests a proper process for changing the permanent council members or which changes the rules regarding who gets to veto what in a way that Saudi Arabia might also qualify for veto-powers one day).
They've never been a permanent member and they even rejected when they were elected to adopt a seat within the UNSC in 2013. They're unlikely to get a seat, because there's no formal application process and no one is pushing for them to have a seat within the UNSC.
Regional powers such as Brazil, Germany, India, Japan, Nigeria, and South Africa have sought to enlarge the Security Council or secure permanent seats of their own. Others have called for France to cede its permanent seat to the European Union in the wake of Brexit, especially after France and Germany decided to share the presidency of the Security Council for two months in 2019. In 2021, Britain announced its support for Germany receiving a permanent seat. And in early 2023, China, France, and Germany called for two permanent seats for Africa on the Security Council. So far, calls for reform have not been heeded, with many countries instead seeking to diversify their diplomatic efforts through increased involvement with coalitions outside of the United Nations, such as the Group of Twenty (G20), a bloc of many of the world’s largest economies. The debate about expansion is often framed as a trade-off between legitimacy and efficacy. Saudi Arabia took the unprecedented step of declining a nonpermanent Security Council seat in 2013, announcing a day after it was elected to a 2014–15 term that it would not serve in the absence of institutional reform.
https://www.cfr.org/backgrounder/un-security-council
https://en.wikipedia.org/wiki/List_of_members_of_the_United_Nations_Security_Council
Question: Is it possible for Saudi Arabia to run for a permanent seat in the UNSC?
Anything is Possible but it's highly unlikely as efforts to expand the United Nations Security council have never included Saudi Arabia. They have rather focused on broadening the geographic representation on the council, and just as importantly recognizing the emerging economic powers who will become most important in the 21st century. Saudi Arabia is neither.
India, Brazil, Germany, and Japan seem to be the major candidates although the short list is larger than that. The argument for giving these countries seats goes to continued relevance of the Security Council itself. The security council only remains relevant by reflecting the diversity of the countries most concerned with continued world order. Germany and Japan have long been global economic leaders and that's why they are on the list. India and Brazil will both be among the most important countries in the 21st century population wise and economically. That is why they are on the list.