Ben Robinson, 14, from Carrickfergus, County Antrim, died after suffering concussion in the match in 2011. He was in several heavy tackles before collapsing near the end of the game.
A coroner found that his death was caused by "second impact syndrome".
He said it was "probably" the first such case in the UK.
The summons, issued at Belfast High Court, also named Carrickfergus Grammar School, Ulster Rugby, the teenager's rugby coach and the match referee.
In a statement, the family said the summons was for "negligence regarding concussive-type injuries" and "their failures in their duty of care owed to Benjamin".
The statement alleges that during the match "there were numerous opportunities for those in charge to notice that Benjamin had sustained several concussive-type head injuries".
It added: "It is the case of Benjamin's parents and family that Carrickfergus Grammar School, the IRFU, the Ulster Branch of the IRFU and World Rugby were in charge of disseminating information with regard to the risks of concussion in rugby and therefore, the coach of the schoolboy rugby team and the referee should have been aware or should have had the means of being aware that concussion is dangerous and could result in death."
The statement also said that the family's legal team had "gathered further evidence to confirm that Benjamin's death was due to second impact syndrome and was preventable".
It added that "proceedings are well advanced".
The family of a teenager who died while playing rugby have issued legal proceedings against the Irish Rugby Football Union (IRFU) and World Rugby.