There is not a definitive answer as to why the ceremonies stop at twelve in The Giver, but there are some reasons toward which we might speculate.  Twelve is a number of completion (much like seven)- there are twelve seasons in a year, Jesus had twelve apostles and there are twelve items in the English dozen to name a few.

Also in the society of the story, puberty seems to have a very strong link to adulthood.  There are many mentionings of "stirrings" which we learn are sexual desires that all members of the society repress with medication.  Twelve is a common age for the onset of puberty and in this story's society, the pills are an unofficial rite of passage--note how Jonas feels proud once he joins those that take the pills.