Will my minor daughter who shoplifted less than $20 get a record?

Unless she can get into a diversion program, she will have a criminal record. The amount taken does not matter, it's the act. If there is no program, a lawyer likely can get a Stipulated Order of Continuance, such that after she has successfully completed her sentence the charge goes away. She will also have to pay a civil fine to the store and make restitution.