You can most likely upgrade your wheelchair with so called "regenerative braking"
that's basically what you have in mind for your kart as well..
No need for an alternator or anything though..
the motor does that just fine (if you use a brushless variant it's even reasonably good at it

)
-well, fine and good is a bit of a stretch tbh-
Just the controller that needs to allow for that sort of thing.
(needs to have a internal charging /rectifying circuit...)
The benefit is rather minimal though for most uses..
As TT540 correctly pointed out.. a downhill portion on the trip is required, and the longer that is, the better of course.
And while RB 'works' even if you just stop at a traffic light,
it's efficiency is.. well minimal (and it'd be WORSE if you add an alternator! the additional drag and weight would in fact
reduce your milage overall)
So yeah, the best thing you could do in fact is upgrade your batteries..
higher density LiPos and/or a bigger pack [higher density means more Ah of capacity w/o adding weight, bigger or more batpacks increase the weight of course

]
So if you're saying you want to build a go kart for fun .. we'll be here to help you with that.
If you "need" a commute to the store, we can help as well, but there are better ways to get you safely there and back than a go kart IMHO.
Especially since AT the store, you very likely will be allowed to recharge a wheelchair;
I doubt they'll be as happy to see you charging a go kart [jus an assumption of course

]
BUT tell us about it.. the weight you need to move (your weight, average shopping etc.), the general road conditions (hills, flats, straights, number of traffic lights, time to get there etc..)
How fast you want to go (and/or allowed to go)
How much money you're able to spend for such thing (about'ish is good enough)
If you're able to work on it yourself, what skills you have (welding, electronic tinkering etc..)
Or if you need to have it made for you (which means a likely unfortunate amount of additional spendings)
And in case you want a kart, what type of kart you desire..
is it okay to sit -well- on a go kart (straight legged on the floor basically) or if you need your knees to be at a more convenient angle to get in and out of the kart...
'sid