We don't do the most of it around here but there are places (sadly I don't have any links right now) that will help blind/partially blind peeps play games by either emulating in giant form, piping out text to go into a text to speech synthesiser, and because I know it will get a groan I am sure you could also pipe it to one of those cool real time braille printers. Mind you with current voice synthesis tech being what it is (very small database of samples means almost indistinguishable to most humans on normal computer) I am actually interested in that one as a means of dubbing games without having to sort through a community of computer nerds to find a whole group with time, good voices, good microphones, some ladies to do the ladies/kids, and the desire to edit/master it all.
That is then usually more for RPGs and other such things and not what you asked for, I mainly mentioning them because if you mostly go for fighting games then this is a whole other world you might be skipping there.
As for the question at hand.
I imagine you could do something like that. Mario Kart might be a bit harder for any kind of useful weapons as many are specifically aim, very tight aim window, obstacles in the world/track and very tight timing. This is not to say something positive could not happen as a result, however compared to other things where you get essentially the same experience as someone else playing the game as is it is going to be a world apart, and I don't know if you can meaningfully play with anybody playing the conventional game either.
My general plan of attack would be the games tend to already have AI to drive around the course. Should not be too hard to flick the player over to AI (there might even be a CPU demo mode type hack already) and thus sort steering and speed. Wrestling away weapon controls from them is comparatively easy.
You might also be able to further reduce options to make things more playable. For instance I don't know how much you know of the games (be it from FAQs, discussions or because you played them before loss of sight) but in it you have green shells and red shells (also blue shells and a few others but let's not got there right now).
Red shells more or less home in on people but can be blocked by obstacles and some other items. Fair enough really. Bonus is they already have an audio cue when they get close.
Green shells fire in a straight line depending upon direction you turn (or backwards to act as a one time shield or frustration effort to a tailgater), can destroy red shells and other items, but bounce a few times after you fire them at barriers. They are also among the most basic and common weapons.
You can probably make something match a vector with a player character (the game itself will likely be doing something like it already, it is usually how FPS games work but guns are generally instant travel for bullets so make a better use of that approach) and send a tone/vibration when it will likely hit.
I say likely as brakes, shield, tight turn, jump, barrier... all ways of avoiding them, on the matter of shields/invincibility there is also the somewhat meta thing of do you make them waste it now or towards the end of the race when it is more valuable). I have also played against 8 year olds that have a mastery of that, and 5 year olds that get a good chunk of the way there.
To that end you could go the other way and restrict weapon choices to those that might do, though it could also make things overpowered as most of the things I would lose are also the most basic and skill capped items. Not to mention you still face the meta problem -- for any of the tracks with a big jump I am saving my lightning (makes everybody small, able to be run over and slow enough to fail big jumps) to do maximum impact, and likewise anybody with a star (few seconds of invincibility from items and a speed boost, as well as knocking out anybody you hit and probably losing their current weapon) is going to know that and save it.
Going further if it is to more or less keep up with some others you might also be able to give yourself a quicker recovery so things don't slow you down as much.
To that end if I am going to do a driving game I am sooner going to either do a rally game (a good one will have a copilot doing the in 200m hard right, immediate soft left routine that they do in the real deal) or something like gran turismo where they have suggested speed curves that you could probably kick to audio cues (high-low-perfect pitch sort of thing) or maybe rumble. Plus they will tend to be more amenable to memorising a track.
Even with all that, and while I am generally not inclined to tell someone how to have their fun, that does all sound like so much boring follow the tone/press when you hear from a hearing test, basically cookie clicker and general boringness. If I find myself at the extreme ends of the blindness scale tomorrow I am not going to try to get Mario Kart back -- I am going straight for the RPGs/text adventures to have something tell me a story and make some decisions or solve some puzzles in it. Or maybe a music/rhythm games (I will assume going blind makes me less tone deaf), suitable puzzle games, or maybe fighting games (some have some nice audio cues and are closer to timings/inputs tests than tests of visual recognition). This regardless of whether it is just me or possibly 7 new mates in the same boat, or me still wanting to be able to play at some level with those I presently play Mario Kart with.