Generally, the types of things that people can exchange for the opportunity to ride are grooming, mucking stalls, and feeding. If you are well-organized and good in mathematics, you might even be able to do some bookkeeping; and babysitting is sometimes attractive to the professional horseperson who has children. Anything that you can do for the professional that is not immoral, unethical, or dangerous is a possibility for a trade-out.
This type of arrangement, however, usually involves trading your work for the opportunity to have lessons, rather than for the opportunity to compete a horse. A professional can give you a group lesson once a week on a school horse without loss of money; but once you’re in the realm of showing, the situation is more difficult. There are shipping costs and stabling fees; so it doesn’t make sense for the professional to let you use one of his or her horses for showing unless the horse’s costs are being covered by a paying client who is competing in other classes at the same show.
Working students often complain that they are so tired by the time they get to ride that they really don’t get much out of the lesson. Although this route is a way to get “free” lessons from a professional, it’s a hard way to go. I think it would only be a good trade-out if you were receiving instruction from a highly-competent professional, so that you could take knowledge with you as part of your compensation, rather than leaving with only memories of time spent on a horse.