Are there any religious orders that only celebrate the Traditional Latin Mass? Or at least favor the Latin Mass?
There are lots of groups that do this.
If you’re interested in being a monk, check out Our Lady of Clear Creek, a Benedictine community.
If you’re more interested in parish work, check out either the Fraternal Society of St. Peter or Institute of Christ the King.
There are loads more.
I saw this list, but it might be misleading. The Dominicans in all provinces in the US celebrate the Novus Ordo in all of their houses. The occasional Mass is said in the Dominican Rite (and it’s becoming more frequent, Deo Gratias!), but the NO is the primary rite.