Trying not to break the bank is going to be difficult due to the type of lens(es) you need.
The camera on the other hand can be almost any reasonable DSLR if your quick with your shutter & have a good eye.
You're gonna need to have a sixth sense for where to point the lens ahead of the karts arriving position.
Panning is an art you need to master as well, search for it you'll find loads of info on photography panning & techniques.
Anyhoo, you will get advise on high frames per second, which can help but if you get the kart in frame & with a high enough shutter speed it should be a keeper.
Of course that depends upon what style of photo you're after, which most likely will change from time to time.
Stock till; motion blur with the kart; or just the back ground; or back ground blur with the kart wheels showing rotation but the kart body is tack sharp; the style of images goes on.
One piece of advise I feel most people agree on is that nearly always buy good glass (lenses) as for the most part once you start investing in glass you don't often sell them.
But the camera body is the item you'll find upgrading from time to time, and of course generally stick to the same make due to the glass you have invested in.
A Canon 60D will give you a crop of 1.6 to help with extra reach for zooming in, a 300mm lens becomes a 480mm.
It has 5.3fps & 1/8000 sec shutter speed, 18m effective pixels, 6400 iso (12800 in high mode) & I've seen them go for less than £300.
As I say the glass is where you need to concentrate on spending wisely, in another thread I have mentioned I have a Sigma (Canon mount) 120mm-300mm f/2.8
I might be telling you something you know, but that lens is pretty fast at f/2.8 & does so all the way through to 300mm, add a 2x convertor & you get a 240-600mm f/5.6 right through to the top end, 960mm on a 1.6 crop camera.
F/5.6 at 600mm effectively 960mm is super fast.
Here's the link to the thread, I have some photos posted there as well,
http://www.talkphotography.co.uk/threads/sigma-150-600mm-sports-lens.569897/#post-6605854
Consider buying a monopod if you buy a large fast lens as they are heavy & usually come with a tripod collar which fits just as easy to a monopod.
And a monopod doesn't get in the way of other spectators like a tripod might normally do, but as a plus goes a long way to help keep things steady & hold the weight!