Making the Student as Person 1 on the Registration Page will allow any additional people to be connected to the student. However, Person 1 must be a Member, so if your club does not have students as Members, you will need to manually adjust the Relationships after registration by selecting each person in People and editing their Relationships. For multiple athletes, you will need to register multiple times.  

If you do not plan on having student accounts for accounting or communication purposes, you can have the applicant list the students' names on the Registration Page in a Custom Field you create in People > Custom Fields > New Custom Field, such as “Athlete Names”.