It's completely possible. It's also a lot harder than you think, Essentially, you have to construct your own playback module that will decode the MP3 streams, mix them together, construct a resulting MP3 stream, and save it to a file.
Honestly, I think PHP is probably not a good language to do that in.
Your best bet is to find someone who has written an MP3 player (and knows how to decode MP3 streams), tell them what you want, and hire them to construct the online mixer you're looking for. Expect to pay somewhere in the range of $3000 - $10,000.