GBA homebrew tended not to have any file readers, and recent years have not furnished us with any. Though as the source is there you could do something.
Beyond that the GBA does not have many outputs -- there is the serial cable, headphone jack and cart slot itself... The serial cable obviously has some things but you will have to make one most likely as buying one is hard these days (not to mention it is slow and you will still need a PC).
If you go up to the DS (and have a DS flash cart as I doubt we will easily get the Omega running in DS mode) you do gain wifi and thus can use FTP or something if you made something to read the Omega, which is possible (there are pieces of homebrew, which we don't recommend using for the EZ4, that will read GBA and DS slots with various DLDI patches).