The 1903 .38 definitely should not be fed .38 Super ammunition. The slide notches at the front of the slide can crack, and cannot be repaired. I do not know if .38 ACP ammo is available today, but it is simple to load your own. No differences between .38 ACP and .38 Super cases. Same is true of the Colt Model 1905 .45 semiauto of very similar design. While today’s .45 ACP ammunition will fit and fire in the 1905, it is too powerful and slide notches will also eventually crack. Only light handloads of .45 ACP should be used in a 1905, never factory or GI.
I once owned a Colt 1903 .38. I do not remember what my handload recipe was, but I worked up charges gradually to a level which was the minimum to reliably function the slide, and I used only lead 9mm bullets.