What other Force-based organisations use lightsabers?
I will be keeping my answer to canon-only examples. The main media that they appeared in are in parentheses.
- The Jedi Order - Given
- The Sith - Also given
- Inquisitorious - Although one could consider the Inquisitorious a Sith group, it was more of a subsect of the Empire than anything. They were a group of Force-sensitive agents trained by Darth Vader to hunt down the remaining members of the Jedi Order after Order 66. At various points between Revenge of the Sith and the third season of Star Wars Rebels, they were all killed, be it by Jedi, rebels, or other unknown means. (Darth Vader: Dark Lord of the Sith, Ahsoka, Star Wars Rebels)
- Ordu Aspectu - This organization was a violence-hating splinter group of the Jedi Order dedicated to prolonging life for all life in the galaxy, in some historical accounts anyway. Other accounts state that the group kidnapped Jedi younglings and sacrificed them to achieve the immortality they desired. Whatever the case, the Ordu Aspectu met their end at the hands of the orthodox Jedi, although they lived on in crystalline form with the archaeologist Chelli Lona Aphra. (Doctor Aphra)
- Knights of Ren - The audience's knowledge of this group is currently limited. It was made up of six former students of Luke Skywalker and was led by Kylo Ren (Ben Solo). They have massacred at least two civilizations, but other than that, we don't know much about them. (The Force Awakens, The Last Jedi)
- Acolytes of the Beyond - This was a group of Dark Side fanatics initially led by Sith cultist/adviser to the Emperor Yupe Tashu. After gaining possession of "Darth Vader's" lightsaber in the wake of the Battle of Endor, they staged a violent revolution of Corellia and attacked New Republic enclaves in various parts of the galaxy. The Force-sensitive nature of this group is up for debate, but many of their numbers have had visions of fallen Sith. (Aftermath trilogy)