If you are serious about getting sponsored I would recommend doing a lot of research to start, check out other eventers' websites and see what they are doing and offering, and start gathering information and put together a basic sponsorship package that you can hand out to potential sponsors. The package should include some information about you, your past experience, your goals, what you are looking for and why you would be a good person to sponsor. People just don't give away stuff to whoever strolls up, the person who presents themselves professionally and actually hands them a package outlining their experiences, goals, and what they are looking for is going to be taken way more seriously than someone wandering around just looking for handouts.
Why exactly do you want to get sponsored? Do you want free stuff? Do you want monetary backing to travel to different competitions? Do you have a promising young horse you are wanting to bring up through the levels? It's not just about money and getting free stuff, a sponsorship is a working business partnership that benefits both parties. So you have to offer something in return, whether it be putting their logo on your trailer, advertising on your tack/clothing/saddle pads, making social appearances, publicly endorsing their products, etc. And then work to fulfill your end of it as well, so whoever it is will be happy with the arrangement and want to continue the sponsorship.
Someone who is well established at their current level and has good results will also be considered more seriously over someone who is newer to the sport and has an inconsistent show record. It's more beneficial for a larger company to sponsor someone who is higher up in the levels, more well known, and doing well because that is more exposure for them. Your biggest hurdle will be trying to convince people why a lower level, grassroots rider such as yourself deserves that support, instead of them giving it to a bigger name. This is where you also have to be a really good salesperson and have some marketing skills! A good salesperson could sell you a wet paper bag to carry your lunch in, so you have to learn how to sell yourself and really focus on your strengths.
It really depends on what you are wanting to do in the long run. It's possible to drum up a little bit of local support if you present yourself decently and put in a some effort. If it's larger sponsors you are talking about, that is harder to break into.