I had heard the term "Orthodox Mennonites" years ago but hadn't thought about it much until my Canadian friend at Fletchington Farms brought them up recently.
It seems every religion has their ultra-orthodox off-shoots. In the Amish religion, it is the Nebraska Amish and the Swartzentrubers. The Nebraska Amish are actually more conservative than the Swartzentrubers. Among the Mennonites, most people assume the Old Order horse and buggy Mennonites are the most conservative.
But, no, there is actually a very small group who call themselves "Orthodox Mennonites" (they are also a horse and buggy group) I don't know a ton about them because I've never come in contact with any.
This is a photo of one of their buggies as seen in Renfrew, Ontario recently. Interestingly, the one thing you can glean from the buggy is that it does display the orange safety triangle. Unless Canadian courts have just mandated it and harshly enforced it and the Mennonites have acquiesced, I find this trait interesting. The most conservative Amish generally don't display the orange triangle even where it is law.
There have even been splits among the Orthodox Mennonites. The Orthodox Mennonites do use meetinghouses (as opposed to Amish who worship in their homes) and the men - unlike other Mennonites - have beards. These conservative bearded Orthodox Mennonites are often called "Gorries" after a town in Ontario they settled near in the 1980s.
The Global Mennonite Anabaptist Enclyclopedia Online (GAMEO) has an excellent entry about the Orthodox Mennonites and their history but, honestly, I got a headache reading it. You need a scorecard, really, because there have been so many rifts and reunions within the Orthodox Mennonite movement.