Sure it's possible. The simple overview of a mod player is it takes the file and generates sound as it goes, a chunk at a time. Without webaudio it does this by creating a wav file that is then loaded and played. The issue is to make one chunk match the other precisely. Webaudio allows you to buffer sounds to be played seamlessly, as well as eliminating the the overhead of creating/loading the sound files by instead letting you write sound data directly. Of course the nitty gritty of it takes a lot more than that and some specific knowledge, not to mention a lot of time to make and optimize. Using an existing js library is dependent on if one can be found and devising a way to use from C2, which is a time consuming task in and of itself.
Converting to ogg is the fastest solution and gives the best performance since sound doesn't need to be generated.