Is it common to restrict access to /var/log/.... ?
Yes
Is it common to allow the owners of websites visibility of their logs?
Yes
Your issue is how to configure this so that Apache can write the logs, users can read their own logs, but not read the logs of other users. That's quite easy using standard permissions.
With a default package setup your httpd must run as root (in order to listen on ports 80 and 443) meaning it can read and write everything (at least the master process can - worker processes setuid to a non-privileged user). But even if it already runs as a non-privileged user this can be resolved using standard filesystem permissions.
Do make sure that the location you choose for the logs is outside the document root otherwise the logs will become an exploitable asset for any attacker.
Also bear in mind that you need to retain or implement log rotation for all these files.
Setting this up from scratch, I would go with....
drwxrwx--- ${USER}:adm /var/www/${USER}/logs/
drwxrwx--- ${USER}:adm /var/www/${USER}/html/ (document root)
drwxrwx--- ${USER}:${USER} /var/www/${USER}/data/ (because its good practice to provide space outside the document root for the users too)
If they insist on running self-modifying code (and you want to let them)....
drwxrwx--- ${USER}:${WEBUID} /var/www/${USER}/html/
(but note that this gives the means for users to not only read others files but also modify them).
It might be simpler from your perspective just to give them each a container and run a load balancer on the hypervisor.
administer an Apache web server for a University CS department.
...so you knew all this already?