Baptism by total immersion is permitted in all Rites of the Catholic Church.
Some diocese still offer the option of full immersion baptisms.
Before going on, I would like to make a distinction in the term Catholic. I believe that this question deals with Catholics of the Roman (Latin) Rite and not Eastern Rite Catholics.
Actually, immersion is a very Catholic way of baptizing, even though it has not been broadly practiced in the Roman Rite. Eastern Catholic Churches have been baptizing by immersion for centuries. It is one of two methods for baptizing that have long been permitted for use even within the Roman Catholic Church.
The increasing popularity for baptizing by immersion is probably related to a liturgical principal favored by the Second Vatican Council: the expressive use of signs. The General Instruction of the Roman Missal says that good signs “will more effectively foster active and full participation and more properly respond to the spiritual needs of the faithful” (20). This
is why we promote such ideas as wearing noble vestments, reading from a beautiful Book of the Gospels, and receiving communion under both kinds. All these signs, when used richly, respond to the spiritual needs of the community. Baptism by Immersion
That said, some dioceses do offer baptism by immersion.
Check out this article: Many Catholics Choose Baptism by Immersion.
For more information see these:
Can Catholics choose to be baptized by immersion?
Holy Mysteries of Initiation
Why do Catholics usually baptize by pouring?
It is not clear when the Roman Catholic Church started pouring water as a method of baptism, nor is it clear why they started using this method. But it is important to understand that the Roman Catholic Church believes baptism saves us regardless of the method employed..
It is hard to get a definite answer on this, but it there may be several factors involved. One being how to baptism the infirm. Another being that Catholic baptisms usually occur near the entrance of the church as a sign of welcoming the newly baptized into the life of the church.
Cyprian advised that no one should be "disturbed because the sick are poured upon or sprinkled when they receive the Lord’s grace" (Letter to a Certain Magnus 69:12 [A.D. 255]). Tertullian described baptism by saying that it is done "with so great simplicity, without pomp, without any considerable novelty of preparation, and finally, without cost, a man is baptized in water, and amid the utterance of some few words, is sprinkled, and then rises again, not much (or not at all) the cleaner" (On Baptism, 2 [A.D. 203]). Obviously, Tertullian did not consider baptism by immersion the only valid form, since he says one is only sprinkled and thus comes up from the water "not much (or not at all) the cleaner." - Baptism: Immersion Only?
If you have ever visited a Catholic Church, you will notice that there is a baptismal font near the entrance. However in centuries gone by the Catholic Churches, baptism by immersion was held in high esteem. In the Early Church, places of worship sometimes had what they call baptisteries and were used for total immersion. Somewhere down the line, the Church moved away from having elaborate pools attached to cathedrals for baptisms to more modest baptismal fonts for baptism by pouring water over the candidates head. The reason for the change may have been a practical one, as a baptismal font can be easily placed in a church.
Baptistery in Pisa, Italy
More information on this subject can be found here: Theological Reasons for Baptistry Shapes.