Try gun shops, gun shows, pawn shops, estate sales, garage sales, for sale ads, place a want ad, check on line auctions.
Those that have been designed to do so. Winchester, S&W and High Standard have all made them.
.22 Winchester auto cartridge is very different than the common .22 L.R.. The .22 Win. Auto cartridge has a larger rim and case diameter & won't interchange. The Aquila cartridge company (Mexico) made a special run of this oddball ammo & it is available in the U.S. from specialty shops for around $12-$15 a box of 50. Email me for cartridge measurements. I have an extra box of this ammo if interested.
22 Long Rifle
Anything labeled 22 magnum, such as a Henry Golden Boy.
Your winchester model 1903 shoots a different ammo than regular .22cal rimfire made today.It is chambered for the .22 Winchester auto rimfire cartridge.This ammo can still be found I believe at Old western scrounger. Try www.old western scrounger.com.I believe they are 12.95 a box for 50 rounds,maybe a bit more.
Several different manufacturers make it (e.g. Remington, Hornady, Winchester), but there is no such thing as "inexpensive" factory loaded 22 Hornet ammo.
On line auctions, want ads, for sale ads, estate sales, garage sales, pawn shops, gun shops, gun stores.
CCI .22 Long Rifle- or any standard .22 Long Rifle ammo (there has been a major shortage of .22 ammo for a year now)
Only if it is chambered for it. If you are not sure, take it to a gunsmith.
Winchester model 04A takes a .22 cal round as it is intended for younger men up to adult
The Model 1903 used a special "proprietary" cartridge called the "22 Winchester Auto". It became obselete in the late 1930s to early 1940s. I did recently here that a special run of the ammo was made, and being carried by a supplier somewhere in the eastern part of the U.S.
There were .22 'shorts' and .22 LR (long rifle) models made. Read your barrel to determine which you have.