How do you keep Flash, Quicktime, Java etc up to date for your non-admin users?

In most corporate environments they would use a Software Management System, that would push out the update that the IT teams had made. The local app would be running with admin privileges so could update the files as needed.

In smaller organisations the overhead of doing this doesn't justify itself. I'm not personally aware that it's possible.

If the software to be installed is delivered in a "wellbehaved" MSI packet even small organizations can push them to clients through Group Policy in Active Directory, without the need for a more advanced software management system. For software that is not properly packaged the repackaging overhead can be terrible. In that case manual install is an option that is more attractive the fewer machines you have.

you can try to push updates semin-automatically using psexec from windows or winexe from linux. in both cases you will need to know local administrator rights, but you do not need to have active directory. [ if you have last one in place you can use automatic software deployment via ad policies ].

programs that you mention [ at least java / flash ] have ways of unattended quiet installation. you can find some hints how to automate upgrades/silent installations of many tools here.