Most kart tracks allow motorcycles to use their track and some hold motorcycle races. There are also a few mini-only race series across the country. And a lot of road racing clubs have mini classes, potentially with lower age limits. Most of the kids in MotoAmerica Jr Cup started racing in some combination of all that with their parents footing the bill.
However, most adult amateur racers that you would meet at a club event probably started as an adult after buying their first bike. If you parents didnt get you into it early and pay, you probably had to wait until you have some disposable income later in life.