The 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?
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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).