May I suggest to look into ... let's say an old Manco 713/715
that'd be a formula 3 style body on a lightweight fun kart chassis.
(it's actually an Indy kart.. but the body is indistinguishable from same era formula 3 cars)
here's one to give you an idea
All you'd need to make the transition would be decals and paint
That way, you could focus on some fixing/engine maintenance and such first;
actually building one is of course possible, but fiberglass mold making is not exactly trivial,
and although an eleven year old COULD do that (with adult advise/supervision)
It's tedious boring and frankly expensive...
I don't see the point if you can get a premade body for cheaper..
welding a matching chassis again is very well possible even for a kid your age (again supervision advised)
Yet again.. those karts are not expensive nowadays... and you could hop in and drive on day three to five (not 50 to 100
)
'sid
PS just found this website:
http://www.miniindy.org/miniindy-information/car-specs/
no price on promokartsusa.com (not a good sign) but... live axle is nice [the manco is a dead axle]
PPS and because of your screen name:
http://www.gokartsusa.com/browseproducts/Mini-Indy-Race-Kart.HTML
(2.4 grand for a proper F1 race kart body.. )
