In Litespeed admin console (cpanel-> plugin-> LiteSpeed Web Server -> Admin Web Console), under Configuration menu, there should be 4 items: Server, Listeners, Virtual Hosts, Virtual Host Templates. Virtual Hosts is the one to look at.
Then your request becomes how to disable access logging in Apache via cPanel since LSWS gets everything from Apache config in this case. This is not a LSWS issue at all. You may check with cPanel support on how to do that.