If a woman has a child out of wedlock, can she baptize the child in church?

Absolutely! Not only can the mother have her child baptized, but she should.

No child should be punished for the circumstances of his/her birth. Obviously, the ideal is that every child be born in a happy, holy Catholic family with a loving father and mother. However, the Church knows that reality is often far different from this. Regardless of the circumstances, though, the Church desires heaven and salvation for all people, no matter what. The salvation of souls is the “supreme law” that governs the Church (see Code of Canon Law 1752).

A child born out of wedlock is still precious in God’s eyes and in the eyes of the Church. The faithful actually have a right to be baptized, and the Church would only be able to refuse baptism under very particular and serious circumstances.