The goal of being able to write to a file and allow a group of users read-only access can be achieved by file ownership and permissions. Let's say all the users that need such access are in 'users' group and you alone have access to the root account on that system. Then do (as root):
chown root:users /var/log/AccessLog
chmod 640 /var/log/AccessLog
From now on, only root will be able to modify the file but all users will be able to read it.
If instead you want to allow the users to write to the file, but need to keep track of who (and when) has modified it, you might use FAM (File Alteration Monitor). Above that, to keep a record of all previous versions of the file, you can put a Version Control System (like GIT for example)) - you can make it cooperate with FAM so that each file modification triggers a commit to the local repository. There probably can be easier ways to do it but nothing comes to my mind (at least nothing that would work on top of a filesystem).