The SSPX was founded as a Pious Union. Under the 83 Code, and upon recognization they would become a Society of Apostolic Life.
Canonically speaking, as well-intentioned as Archbishop Lefebvre was, the SSPX 'Third Order' does not and will never exist, it cannot exist! A religious order (not congregation or society) needs to have a First and Second Order in order to have a Third Order. Only the great, major orders have them. The following are the only existing Third Orders:
Augustinian
Benedictine (actually not Third Order properly speaking, but rather Oblates)
Carmelite
Discalced Carmelite
Dominican
Franciscan
Mercedarian
Minim
Norbertine
Servite
Trinitarian
This is why many pre-Vatican II congregations had associated confraternities and societies but not Third Orders. For example the Redemptorists have the Confraternity of Our Mother of Perpetual Help and St. Alphonsus. The Jesuits the Apostleship of Prayer and the Happy Death Society.
As far as Motu Proprio groups (Societies of Apostolic Life), the FSSP has the Confraternity of St. Peter. The Institute of Christ the King has the Society of the Sacred Heart. The Fraternity of St. Vincent Ferrer has an associated Confraternity of the Rosary. On the sedevacantist side, the CMRI has the Confraternity of Mary Immaculate Queen.