The Avrille Third Order was created by them, it is not properly speaking the Dominican Third Order. Why? For good reasons...
They have a re-written, different Rule than the pre-Vatican II Dominican tertiary rule.
They call themselves the Order of Penance of Saint Dominic (compared to the Third Order of Saint Dominic) so at least they admit it's a new creation. They tried basing it on the ancient Third Order...
In their profession formula, they profess to the Prior of Avrille and not the Master General. Even with the situation in the Church, supplied jurisdiction, etc. this is wrong.
With the Capuchins however (although the SSPX controls it in the US) the Rule, etc. has remained the same as before Vatican II. As well as the SSPX-associated Carmelite Third Order and the Benedictine Oblates associated with Silver City, Bellaigue, etc.
You can't have a third order unless you have a first order... that is the problem... and neither the SSPX or the resistance have a first order. The only legitimate third orders in existence are Dominican, Benedictine, Carmelite and Franciscan.
Sure, it's nice to have a group of lay people that say extra prayers etc... but you can't just organize this and call them a "third order". The first order of each of the 4 orders listed above are the priests of those orders. The SSPX is not an "order" per se but a "society" of priests.
I would suggest that anyone that wants to be in the third order, to join one of the 4 existing third orders, some of which have been in existence for more than 500 years and have the full approval of the church. Despite the crisis in the church, you can still find traditional versions of each. The Dominicans of Auvrille, France come to mind.