Depending on the athlete, there are different channels to becoming a professional sprinter. I would assume that by professional sprinter, you mean a sprinter that I sponsored by a larger organization. In reality, some would consider a runner to be a professional sprinter by winning prizes that meet a certain monetary value.
The runner could be recruited from a college to be sponsored by a company. The athlete could apply or be recruited onto a running team that already has a sponsor, or a sprinter could even have their own coach and participate in official races on their own (see USA Track and Field as a reference to local, national, and world level meets).
Regardless of which route they take, from what I have seen/done, you pay (or your team pays) to participate in official races. As you make faster times, you may qualify or be invited to compete in higher level races.
Usain Bolt, from what I have read, he has been on the radar since at least the age of 14. He started in the competitive races of his region and moved to world level races. I am not sure exactly sure at what point he was considered professional.
EDIT (in response to comments below):
"USA Track and Field Athletes Bios" will give you you dozens of athletes, the path they took, times, races. In these you will see the names of some clubs, teams, universities, and coaches that the athletes have trained with.