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Nice question, I have done some research about the hacker scene and computing tech in the GDR and in consequence also other Comecon countries.
Very, very unlikely that there ever was an official CPSU website.
The communist party CPSU was dissolved on 29. August 1991 and banned on 6. November 1991.
The SU ccTLD (along with the domains dd for the GDR, cs for Chechoslovakia and yu for Yugoslavia) was assigned on 19th September 1990 and registration was handed over to RIPN.su.
However, the first WWW-Site ever (http://info.cern.ch/hypertext/WWW/TheProject.html) was only published on 6. August 1991 at the European Organization for Nuclear Research CERN. Three weeks before the CPSU collapsed and two weeks before the coup d'état attempt. So there simply was almost no time for an official CPSU.su website.
The required software for the WWW was put in the public domain by the CERN on 30 April 1993[1]. Almost 2 years after the CPSU was dissolved.
Before the rise of the WWW in the mid to late 1990s the Internet was used for email and mailinglists and for discussions in the Usenet. The Usenet was created in 1979 and became a big part of the Internet in 1986, with the publication of the RFC977[2] defining the Network News Transfer Protocol. It was a hierarchy of newsgroups ordered by topics. Comparable to Reddit, but only text/ASCII based.
Back on 1st April 1984, Konstantin Chernenko announced in the eunet.politics group that the Kremlin connected it's VAX (One of the larger computers built by Digital Equipment Corporation and fundamental in the birth and spread of the Internet) to the Usenet. This way, the communist party and the KGB would be able to participate in the Usenet/Internet [3][4].
However, as you already may have realised, it was an April's fools joke made by Piet Beertema, who 4 years later would link the Netherlands into the NSFnet. This joke was known as the Kremvax hoax and caused confusion, when a real Kremvax showed up in 1991 and was active during the coup in Moscow. This incident made it to the Jargon File, the hackers dictionary and a great source and part of early internet/hacker culture. And back in the day it was forbidden to export western computer technology to the USSR and other Comecon countries, so the USSR and GDR had to get their hands on western computers by covert operations and would not announce that they own a DEC VAX on the Usenet.
The role of the Usenet in the 1991 coup has been further described in [6].
As of today, https://cpsu.su/ exists as a part of the »SOVNET initiative to recreate the Soviet segment of the Internet«.
http://sovnet.su/ gives some information about the Internet/Usenet in 1989-1991 USSR/Russia. Use the google translate button to read it in English. According to them, ca. 60 .su-Domains were set up and used, mostly by research institutes and universities, like msu.su for the Moscow State University.
To sum it up: It's very, very unlikely that an official CPSU website existed, simply because the WWW was just being created as the CPSU ceased to exist. The only authoritative answer (pun intended) could give the domain registrar/soviet/russian network operations center. But back in the 1980s/early 1990s a lot of files were not archived, so I doubt they still have a backup of there DNS records when they activated cpsu.su
[1] https://home.web.cern.ch/science/computing/birth-web
[2] https://tools.ietf.org/html/rfc977
[3] https://groups.google.com/g/eunet.politics/c/_WKpffStBPc/m/JaaDBecKCM8J
[5] http://ftp.informatik.rwth-aachen.de/jargon300/kremvax.html