I think you can do this with a workflow. As I don't use Cases with Clarizen, let me ask a few question to help me understand:
- When you say "Submitted", do you mean "Created", or is this related to some sort of state (in this case, what field is modified?)?
- When you say "adding of team members", can you tell me what field is used? Assigned to, Groups, something else?
- Do you add directly a group, or one or several users?
I might be able to help you with these details.