I exercise horses, and no, I have not experienced discrimination. I ride primarily English, however, where it's more of a "we don't care or care to know" attitude on people's private lives. If you rodeo then you may have a more difficult time, and I would suggest that you remain stealth for your own safety. While many cowboys will accept you there are some who may be violently opposed (but not as violently opposed as they are to gay men--it's tough being a gay cowboy), plus you may face ridicule. Though if you have mad skills then you will likely get respect for that. If you ride hunter/jumper, eventing, dressage, saddle seat, or some other English based style of riding then you will likely have no more than the usual experience with people being surprised then getting over it with a few being freaked out. American men at professional levels of these events are used to being mocked and called gay for their sport so they are highly unlikely to mock someone else. Sponsors just care whether or not you win. So no, I would not expect it to ruin your chances of having a career. You might want to keep it on the down low, however, not out of shame but rather because sponsors are not interested in your personal life being the talk of the town--they want people talking about their horses. Notice that you hear almost nothing about famous riders' personal lives in magazines like the Chronicle of the Horse. That's because in the sport it is not considered appropriate to go around blabbing about your marriage or divorce or sexuality or kids or gender issues--if it isn't about horses then it isn't a topic for the showgrounds. Professional riders are riders first, everything else second. So I would just come out then let it go and not try and drive it home that you are a trans rider every five seconds as horse people only care that you are a rider.
When I came out, people at my barn didn't even blink an eye. I'm not sure the my gave a rip as long as I was still riding the horses, hauling the hay, and scooping the manure. People at shows might whisper (because English riders love their gossip), but I doubt it's as interesting to them as which of their rivals is dating which of the grooms. There are no gender divisions in riding, so you don't have to deal with regulations on what you participate in. Even clothing is no different. Well, my men's breeches have back pockets and I can now wear a traditional tie rather than a mandarin or stock tie under my jacket at shows without attracting odd looks, but that's it. Overall, I just announced my move then never talked about it again and everyone was cool with that. No one is interested in my gender transition--they just want to know where I got my saddle.
You are correct that it is a conservative (and "tough person") sport, but I really would not call the culture "macho"--girls are expected to be tough, too, so it's not really gender related there. Rodeo, on the other hand, is VERY macho, as is horse racing (there are very few female jockeys and I don't believe there has even been a female All Around Cowboy--too few ride bulls--so you might be BETTER off as a trans man in rodeo or track than a female would be, especially if you don't shout it from the rooftops.)
Hope this helps!