Although none of the SSPX chapels I've ever been to have done this, it would not be completely out of place if the Sanctus was sung by the laity.
This is what a priest wrote concerning this:
"This is all partly due to a misunderstanding of "tradition" or "custom". The custom of kneeling during the Sanctus and Agnus Dei come from the Low Mass, which originally was a private Mass in which people could piously attend. This is certainly a very good thing. However, that is not the original custom liturgically, but comes from a time when the Low Mass displaced the High Mass as the norm in many, though not all, places. The rule is that one stands during the times one sings, and the Sanctus and Agnus Dei are meant to be sung by the people alternating with the Choir. You should notice that the choir never kneels during either, and that is because they are singing. When the people quit singing their parts, then they stopped kneeling, logically enough. But that is not "traditional" in any real sense of that word. It is interesting that the Council of Nicea forbids all kneeling on Sunday- not to insist on that, but that it shows that the original customs were quite other. One should not confuse and abuse with a legitimate custom. For Low Masses it is great, since the people are not exercising any liturgical function."
And this is what someone else posted in response:
"It has nothing to do with a hybrid Mass ...! The rule is that, at Mass, you don't sing on your knees, and I was taught this nearly 30 years ago by the traditional priest who received me into the Church. I think that in the countries influenced by the Liturgical Movement - France, Germany, Switzerland - the laity follow this, and of course the seminarians are trained to do this at Econe. But in England and America people tend not to, being more influenced by Low Mass customs, so there's usually trouble when Continental priests try to introduce it. As happened with a French priest in the US a few years ago, I remember!"