How can I install or update software with yum in CentOS 6 when CentOS 6 stopped its lifecycle?

All things considered about security and it being EOL, you can still make yum usable in CentOS 6:

curl https://www.getpagespeed.com/files/centos6-eol.repo --output /etc/yum.repos.d/CentOS-Base.repo

Or, update repo configuration manually.

Officially there is nothing you can do with CentOS 6 - it's dead and buried - there will be no more updates from the CentOS project. From the CentOS-devel mailing list on 9th Nov 2020

All,

This is a friendly reminder.

CentOS 6.10 will EOL at the end of November 2020.

During the first week in December 2020, the 6.10 directory will move to vault.centos.org

Packages will still be available at:

http://vault.centos.org/centos/6.10/

However, once moved, there will be no more updates pushed to vault.centos.org. Therefore, security issues will no longer be fixed, etc.

You should take the rest of the month to either move to a newer versoin of CentOS Linux ... or to procure Extended el6 support from Red Hat (EUS RHEL 6).

your best course of action is to upgrade to a supported OS version.

From the CentOS forums:

Red Hat have pulled the plug on RHEL 6.x as of Nov 30th 2020 and as a result CentOS 6 is now a dead version. The online yum repos for CentOS 6 have been archived to vault.centos.org and there will be no more updates to it at all, ever.