Does HC10 support Webalizer stats?

Yes, Webalizer Stats Server is shipped alongwith HC10 installer for both IIS and Apache WebServers in order to enable these on your websites. You can have a look into supported softwares from the following link:
https://portal.hostingcontroller.com/kb/a76/supported-third-party-softwares-with-hc.aspx
 
If you need further help for configuring it in your HC panel, you may contact support department by opening a ticket with support@hostingcontroller.com