Saying "like any other stored program computer" makes it sound like there were a bunch of them, but as I understand it there was really only ONE up and running at that time in 1948 - IBM's SSEC. The stored program EDVAC, whose architecture Von Neumann had formalized, was still not yet running, and Von Neumann personally advised on adapting ENIAC to this mode of operation.

As I understand it, even after adaptation to "stored program" mode, the program was still configured rather than loaded and stored, but now configuration was done using the function table dials (essentially a dedicated dial for each program step) rather than patch cords, and was a lot more convenient. Punched cards could be used for reading/writing data, but not for programs.

It seems there was no BIOS because I/O was so primitive. Per configuration a punched card would be read into the "constant transmitter" unit, then one of the read numbers could be copied into one of the 20 accumulators. Or you could do the same in reverse to output data from the accumulators.