You’ll have to ask your priest or RCIA director how they do it at your parish.
The Missal (the book that outlines what is done during Mass) only says “After the Baptism, the Priest anoints the infants with chrism. A white garment is given to each, whether adults or children.” Nothing is specified about what the white garment looks like, how it is worn, or when it is removed.
Most places I’ve seen, the people do keep them on, but it’s possible your parish does it differently.