Hi Carly, I think the question is not so much which camera, but which lens.
If you want fast focusing and good performance in low light, you will need to buy something better than the kit lens which comes with most DSLRs as standard.
It also depends how much you want to zoom-in on the horses when they're at a distance. Do you need to zoom-in a lot, or just a little?
If you need something that can do wide angle AND zoom-in a lot, then you'll either need 2 lenses, or a superzoom like the Nikkor 18-200mm VR - and this will only work on a Nikon body.
If you don't need to zoom-in too much and good performance in low light is more important, then you need a lens with as small an 'f-number' as possible, although these are expensive. A lens which has f2.8 will be much better than one at, say, f3.5 or f4 or f5.6 when you're in darker areas...
A great option for Canon bodies is the EF-S 17-55mm f2.8...
Similar lenses exist for other cameras though, so I think the first step is to try and work out how powerful a zoom you need, whether you're willing to carry two lenses and just how dark some of the lighting conditions can be.
If you know someone with a lens range that you like or know, this will also be useful - so you can say - a friend has an 18-55mm and it doesn't zoom-in enough, or it doesn't get wide enough for example...