I will take the opposite stance.
You could do it with conventional hacking approaches. The effort required however would probably be considerable.
The Wii is the last of the more embedded/single task approaches to systems development, in more Wii centric terms this means no great underlying OS running in the background (aka why every game does not support it but every game happily chats on the xbox 360) and what things are graciously provided by Nintendo are kicked to either libraries or IOS modules (why you have to multiple custom ones installed and different games need different ones) with Wii Speak being one of the latter.
If you are playing Wii mario kart online today then you are also going to be technically inclined at least a little bit and thus it becomes 1000 times nicer to kick it to a phone, tablet, laptop... running some agreed protocol*. Everybody running it would also probably need to install a hacked version themselves lest they be left out of the fun (I seriously doubt you could piggyback it onto existing code, though outside chance you get a custom IOS installed that can do the deed).
*for something like the xlink kai VPN setup for playing system link games on the original xbox an external management protocol was basically required (think giant lobby/looking for game setup) and thus you inherently had some level of communication possible as part of it.
Anyway the general workflow would be to find the multicast server for the Wii online mode (probably fairly basic if the save wifi projects are anything to go by), figure out how to get mic in, encode it in a reasonable to broadcast manner (not necessarily that easy to do in real time on top of an existing game, though the IOS might provide) and either sideband if wasted space is available (it is how SMS text on phones works) or add additional packets that might get ignored on stock but work on modded to decode it back to voice on the relevant parties.
If I was going to do this I would first probably make a text chat work as it is going to be all the same things.
As far as source code. For the Wii I imagine they still have it around somewhere -- lost source is more of a 32 bit to PS2 era problem, even more so among the oodles of bankrupt third parties but first are far from immune, though later examples certainly exist and the legacy of those eras also still taints things (version control in game companies if you read CVs and job postings of the time was not even that of the rickety open source world and far far worse). I am not sure offhand what it would have been coded in but probably C++ which is going to be a fair while before sensible decompilations happen there.