Hat tip to @BrianZ who summarized it well; this is a research opportunity. There are a couple of sources that might serve as seeds for research, but there does not seem to be a common agreement about the right to bear arms in English law.
The best citation I've found is the following:
There was no ancient political or legal precedent for the right to arms. The Ancient Constitution did not include it; it was neither in Magna Charta 1215 nor in the Petition of Right 1628. No early English government would have considered giving the individual such a right. Through the old militia laws —Henry II’s Assize of Arms (1181) and Edward I’s Statute of Winchester (1285)— early governments had imposed a responsibility on subjects, according to their income, to be prepared. Schwoerer, Lois G. (2000). "Tö Hold And Bear Arm: The English Perspective" (PDF). Chicago-Kent Law Review.
I found that source after consulting Kevin Stroud, a lawyer, historian, and creator of the (excellent) History of English podcast who admitted that this was not his particular area of expertise, but suggested that:
. . . many people trace the origin of that right to a proclamation issued by Henry II in 1181 called the “Assize of Arms of 1181.” It wasn’t so much a “right” to keep and bear arms as it was an “obligation” to maintain such weapons.
The assize reads in part,
Moreover, every free layman who possesses chattels or rents to the value of 16m. shall have a shirt of mail, a helmet, a shield, and a lance; and every free layman possessing chattels or rents to the value of 10m. shall have a hauberk, an iron cap, and a lance. Assize
The Assize establishes arms as an obligation of a full citizen and prohibits the right to Jews and others. The requirement extends down to the lowest born freeman.
Notes on historiography
My first temptation when I read the question was to begin analysis from a modern, American point of view; I could begin there and look backwards for sources supporting a connection between the English "right to bear arms" and the corresponding right expressed in the US Bill of Rights.
There are some obvious problems with that, and I'd just heard a podcast that referenced the definition of churl including the right to bear arms. It might be more successful to begin with an early source and work forward to OP's quote from the English Bill of Rights.