It has been suggested that offering green cards to Russians with STEM degrees can be an effective sanction against Putin. What would the political process for that be? Which political body in the USA could make such a decision?
Since the collapse of the Soviet Union in 1991, millions of "Russians" have emigrated to western countries, mostly to the United States (see here) and Israel (Jewish Russians, see 1990s post-Soviet Aliyah), and to lesser extent to Europe and Australia. I put "Russians" in quotation marks, since westerners has been for decades calling Russian everyone coming from the USSR, even if they were Ukrainians, Georgians, Latvians, etc.
This immigration included many skilled immigrants, notably scientists - that is not simply people with advanced University degrees, but some world-known E.g., Alexei Abrikosov received his Nobel Price as an American for the work that he had done many years ago while working in Soviet Union. Garry Kasparov is another high profile example of a former "Russian" living in the US.
With Russian borders open (unlike during the Soviet era), anyone wishing to develop their research or engineering career in US or Europe was free to do so, as long as they had enough talent to compete with the locals and those coming from elsewhere (India and China are two notable sources of qualified Western manpower).
Thus, what we are facing now is rather a possibility of a reverse trend: in its desire to hit "Russians" some western countries crack on scholarships and visas for Russian citizens (e.g., Belgium) - ironically hitting the very people who are more pro-western. This is not a new phenomenon - e.g., Japanese Americans were famously interned in concentration camps during the World War 2 (see Internment of Japanese Americans). Similarly, Jewish and political refugees from Germany wandering in Europe were interred in camps at the beginning of the war as enemy citizens, and later often delivered to Germany, when the host country was overrun by the Wehrmacht.
What would the political process for that be?
Congress could amend the Immigration and Nationality Act, which establishes the criteria for admitting new immigrants to the US.
Which political body in the USA could make such a decision?
Congress, through the normal procedure for enacting statutes (which means that the president must assent or that congress must override the veto). There are various decisions vested in the president or the Secretary of Homeland Security, but they do not have the authority to create new categories of immigrants, and they do not have the authority to increase the number of immigrants eligible for existing categories. The president can increase the refugee quota but cannot make refugee status contingent on educational credentials.