Does the Catholic Church permit girls to serve as altar servers?
The (230 §2)Code of Canon Law permits the use of altar girls:
All lay persons can also perform the functions of commentator or cantor, or other functions, according to the norm of law.
Understandably, this canon might not seem to specifically address your question and back in the 1990s others inquired further into this. It was determined at that time that the canon does, indeed, permit the use of girl altar servers at the discretion of the bishop.
In 2004 the Congregation for Divine Worship and the Discipline of the Sacraments reiterated this determination in (47)Redemptionis Sacramentum:
Girls or women may also be admitted to this service of the altar, at the discretion of the diocesan Bishop and in observance of the established norms.