Can the SQL Server 2012 service be running while applying CU 8 to SP 3? I want to apply CU 8 to SQL Server 2012 sp3 and was hoping to not have to ask users to logoff the system while applying since we have 24 x 7 users on this server. Then I assume if the server does require a reboot it will tell me so or is a reboot something I should do as a best practice?
You will definitely need to have users log off during the patch application; in theory, the service will be available for a while during the process, but there's no way to tell when the service will be stopped, and user activity could impact the patch application. 

Be safe, plan for a downtime, apply the patch. Make sure you have proper backups and recovery plans in place in case of problems.