Is there a way for a person to add themselves to a Google Group?

Yes, depending on the Group settings. By default, people can become group members only if invited to join by a group manager. However, if the "Join the group" permissions (in Permissions -> Basic Permissions) are set to "anyone in the organization" or "anyone in the organization can ask", Connect account holders can join, or ask to join, the group. For more detailed information about joining a group, including by non-Connect accounts, see this help page.