According to the Catholic Church, does any child have the "right" to be baptized? The Church, echoing the affirmation of the LORD himself, teaches that  Baptism is necessary for salvation.1

Furthermore, parents are obliged to take care that infants are baptized in the first few weeks; as soon as possible after the birth or even before it, they are to go to the pastor to request the sacrament for their child and to be prepared properly for it.2

And CCC 1261 ends by saying, 'All the more urgent is the Church's call not to prevent little children coming to Christ through the gift of holy Baptism.'

With the backdrop of all these Church teachings and practice, and noting point 52. of Synod 2014 of "Relatio post disceptationem"


  
  Without denying the moral problems associated with homosexual unions, there are instances where mutual assistance to the point of
  sacrifice is a valuable support in the life of these persons.
  Furthermore, the Church pays special attention to [...] children who
  live with same-sex couples and stresses that the needs and rights of
  the little ones must always be given priority.
  


does the Catholic Church teach that any child, or all children have the "right" to be baptized?

Does the Church teach that anyone has the "right" to be baptized?

If the answer is in the affirmative, who granted those rights?

Please note "Rights" are to be understood as the Church understand them.

1. cf. VI. THE NECESSITY OF BAPTISM | Catechism of the Catholic Church, 1257-1261.
2. cf. Code of Canon Law Cann. 867 and CCC 1250-1252.



Right (Definition noun 18. | Dictionary.com): a just claim or title, whether legal, prescriptive, or moral: "You have a right to say what you please."



Please see also: According to Scripture, does anyone born in the world have the right to be baptized?
It is always helpful and useful to start with definitions

From the "PENNY CATECHISM" under Grace and Prayer


  139 What is Grace?
  Grace is a supernatural gift of God, freely bestowed upon us for our sanctification and salvation.
  
  140 How must we obtain God's grace?
  We must obtain God's grace chiefly by prayer and the holy Sacraments.


And From the "PENNY CATECHISM" under Baptism


  256 What is Baptism?
  Baptism is a Sacrament which cleanses us from original sin, makes us Christians, children of God, and members
  of the Church.


Answering

It readily follows since Baptism is a Sacrament, a means by which we obtain God's grace,  a supernatural gift1 of God, freely bestowed upon us for our sanctification and salvation, it is not a right but a gift freely given to be freely and worthily received.

1. cf. gift definition.