I found a bunch of info on these at the National Archives.
Basically back pre WWI, shooting was actually a really popular sport. There were shooting competiions everywhere and nearly every town had one or several shooting clubs that were backed and controlled by the National Rifle Association. Back when the NRA was actually about furthering shooting and not politics like they are now.
Since shooting was actually considered a huge sport at that time, all they had to do was I believe have a minimum number of members (I think 10), fill out paperwork as a shooting club to the NRA, and they could purchase these rifles thru the club to be used in shooting competitions.
The rifles were technically the property of the shooting club, and not the individual. That was the way they could go around the laws of the time. Though you also could buy one if you were a lifetime member of the NRA.
The rifle sales policies changed about 1919 on this and individuals could now buy them direct, without being a member of a NRA club or lifetime member. The 1919 National Matches were the first M1903's sold directly to civilians who competed in the 1919 NM's.
I live near Camp Perry and these early NRA sales or club rifles pop up a lot in this area. I think just because I'm by Perry. But I see them a lot.
They actually sold a pretty good amount of these rifles pre WWI. I think a lot are still undiscovered in older guys closets and they don't know what they have.