Can an organization within Tribal TANF be sued? Or, does tribal sovereignty give the company immunity?

The TANF program is part of tribal government and it therefore depends on whether the tribe has waived its sovereign immunity in some fashion. Tribes often waive sovereign immunity to the limits of any insurance they may have purchased. So if it's a slip and fall type of injury, you may be able to submit an insurance claim. If the claim is more about how you have been treated, then you have to exhaust the grievance procedure. If you have exhausted and are dissatisfied, you can inquire whether the tribe has waived sovereign immunity so as to challenge the outcome in tribal, state or federal courts.