The .22 Remington Auto has a larger case diameter than the .22 LR (.245" vs .225"), so you can't rechamber the old gun for .22 LR.
I am afraid your friend has a wall hanger, unless he can find a few rounds of ammo at collector's prices. Old Western Scrounger, often a source for older ammo, has the .22 Winchester Auto, but not the Remington.
FWIW, the reason both Remington and Winchester brought out special cartridges with their semi-auto rifles was that at the time, .22 LR manufacture was not well standardized and many ammo makers did not have good quality control. So the only way the arms companies could be sure of providing the quality ammo necessary for their new rifles was to make it themselves. The .22 Winchester Auto is the comparable, but not interchangeable, cartridge for their Model 1903.
Jim