The traditional Latin Mass (TLM) was the primary Western liturgy in use prior to 1969 when Pope Paul VI established the Novus Ordo liturgy. The Society of St. Pius X (SSPX) is a priestly society of apostolic life (sort of like a religious order but without public vows) whose members service chapels that celebrate the TLM. So while the SSPX uses the TLM, not everyone who uses the TLM belongs to the SSPX (and strictly speaking, only priests are really “members,” not the laity who attend their Masses) or even supports it. There are other priestly orders who use the TLM, such as the Fraternity of St. Peter or the Institute of Christ the King, and there are also diocesan priests who do so as well.
The SSPX is in communion with the Church, but there is some controversy about their canonical situation. However, the fact that Pope Francis granted them faculties for confession and marriage shows that they are in communion with Rome, because people who are not in communion with the Church cannot possess habitual faculties. Therefore, you may attend SSPX Masses and other approved TLMs while still being in communion with the Church.
However, there are some TLMs you need to watch out for. For example, there is a group called “the Resistance” who splintered from the SSPX because some people believed that the SSPX was becoming “too liberal” or “too close to Rome.” The Resistance isn’t an organized, established society, and it has no canonical status in the Church. The priests are usually not incardinated anywhere, which means they operate in defiance of divine law which requires that priests possess some kind of superior. On a similar note, there are also some priests who celebrate the TLM who claim to be “independent” priests. Anyone claiming to be “independent” as a priest is lying to you, and you should avoid his Masses.
Members of the SSPX and other approved societies/orders, along with diocesan priests, do not hold a sedevacantist position. Even most Resistance or “independent” priests do not hold this position, even if they act in schismatic ways. Sedevacantism, a term which comes from the Latin term sede vacante referring to the Apostolic See being vacant, is the belief that there is currently no pope; some sedevacantists believe that the last valid pope was Pius XII, who died in 1958, and this is usually the view people refer to when they discuss sedevacantism. More recently, some people have argued that Pope Francis was not a real pope because Benedict XVI never validly resigned (a ridiculous argument) or due to alleged “rigging” of the 2013 Conclave by certain Cardinals (which wouldn’t actually invalidate the election).
The TLM is considered to be under attack because in 2021, Pope Francis issued Traditionis custodes, which restricted the celebration of the TLM. As a result, many churches using the TLM have been threatened with closure.
Coming back to the question in your title, I guess this would depend on what it means to “reject” the Novus Ordo. Some people, mainly sedevacantists, argue that the Novus Ordo is invalid (which means that no consecration actually occurs) and not a Mass at all. The SSPX’s official position is that it is a valid Mass but can be a danger to the faith because it does not express theology as clearly as the TLM. However, there is a range of views in terms of practical application: some priests leave it to the discretion of the individual as to whether he should attend the Novus Ordo, while others say you should never attend the Novus Ordo and stay home if you don’t have a TLM to attend.
I’ve spoken in some generalities here, but please remember that people usually aren’t quite so black and white. I’ve tried to avoid inserting in my personal biases.