The Mashpee Wampanoag Tribe is a federally recognized sovereign nation. Sovereign immunity applies to sovereign nations, which means that the tribe (and those acting in their official capacity on behalf of the tribe) may not be sued in Tribal Court unless the Tribe voluntarily waives or surrenders immunity in that particular case or subject-matter. If one exists, a waiver of immunity is typically found in the sovereignty's Constitution or in ordinances and rules implemented by the tribe's governing body (the Tribal Council in the case of the Mashpee Wampanoag Tribe).
The Mashpee Wampanoag Tribal Court can hear civil actions involving lawsuits against the Tribe (provided there is a waiver of tribal sovereign immunity) or cases involving individual tribal members who agree to the jurisdiction of the Tribal Court. The Court can also hear contract actions where the contract specifies Tribal Court as the sole remedy in the event of dispute. Currently, the Tribal Court does not hear criminal cases.
If you are interested in learning more about tribal sovereign immunity, the National Congress of American Indians offers information on various areas of tribal government in their publication, Indian Nations in the United States.