I don't think there was a "native format" per se, you could write any code you wanted that controlled the sound chip. The wave data samples were in a 4-bit ADPCM format - see section 3.2 of the SNES development manual.
The way the SNES audio worked is that games uploaded sample data (in the special format) along with code for the SPC700 microprocessor. The .spc file is basically a dump of the complete state of the SPC700 and the other sound chips (RAM, registers etc).
Today there are some homebrew projects that take MOD Tracker music and convert it into a format for a SNES SPC program to use. I think an earlier one was called XM2SNES which took XM tracker files. As Jonwil said the .SPC format SNES music is often distributed in is actually a "save state" of the SPC unit. Very few games load data to the SPC while audio is playing so this works really well to distribute SNES music. Clay Fighter and Earthworm Jim 2 are fairly well known for using techniques that break this strategy as they require the SNES CPU to be actively feeding them new data as the sound plays.