Was knowledge of the pronunciation of the Tetragrammaton a component of the "deposit of faith" entrusted to the apostles and their successors? According to the doctrine of apostolic succession, the Lord Jesus Christ entrusted and delivered the faith (i.e., the depositum fidei) to his apostles whom later entrusted and delivered it to their successors, and so forth.

According to Catholics,


Was the knowledge of the pronunciation of the Tetragrammaton a component of the "deposit of faith" entrusted to the apostles and their successors? 
If so, what is the correct pronunciation?
If not, why was the name of God (i.e., the pronunciation of the Tetragrammaton) not entrusted to the apostles and their successors?
The answers to the O.P.'s questions are simple:


No, the pronunciation of a word would not enter in any meaningful way into the contents of the faith. Faith has to do with God and those truths revealed by Him. (See Catechism of the Catholic Church no. 156.) The revelation of God as “I Am Who Am” (see Ex. 3:14), which is deeply linked to the Tetragrammaton, does enter into the deposit of faith, but not the way those concepts are expressed linguistically.
Hence, the question of the correct pronunciation is for linguists, and the Church takes no official position. (As a practical matter, there does not seem to be any way to find out what the correct vowel pattern is, since vowel marks were added to Hebrew only after the Tetragrammaton had long ceased to be pronounced.)
The pronunciation of the Tetragrammaton was not entrusted to the Apostles precisely because it is a linguistic question, not a theological one. It is not essential (or even important) for understanding the mysteries revealed by God.