I would definitely talk to the gym owner of whichever gym you are thinking of. If they know there is a financial need, they may find ways for you as a parent to pay for your child’s classes/fees through working (administration/secretary work, or putting in time helping clean the gym- because that takes way more time than people realize- are two of the most common ways. I’m a coach at my daughter’s gym, and I know a few other rec teachers other have daughters who are on team and we exchange hours for tuition). Definitely talk to the gym owners
To my knowledge, there are no scholarships or free money to do gymnastics at a private gym, particularly if just starting out. As stated above, some gyms allow some kind of barter-services in exchange for classes.