You can get a sponsor if you can bring a company's name in the spotlight. They need to benefit from it, it's what they pay a sponsorship for. So when you compete at a high level, a sponsor can pay to have their logo on your lorry, company name in your horses competition name, or they provide your clothes with their logo on it. Also social media is getting more and more important. Even if you don't compete in big shows but you have 100.000+ followers on Instagram for example, it gets interesting for a company to get their product into your pictures or videos.
I'm pretty sure you have to have compete and be at be a certain show leave to become a sponsor although for kids pony magazines you may not have to. but I could be wrong