Some say we don’t make a brocha because it is not possible to do the whole mitzvah alone; our teshuvah is only complete if it is accepted by Hashem. It is similar to the reason we don’t make a brocha on giving tzedakah; one cannot give tzedakah alone there must be a receiver as well.
But isn’t Hashem always waiting and ready to accept our teshuvah? Isn’t it true that Hashem accepts our teshuvah without a doubt? Isn’t that the reason we make a brocha with Hashem’s name when we say “melech mochail v’sola’ach” (the King who forgives and pardons)?
Therefore, others say we don’t make a brocha on teshuvah because it’s a mitzvah haboah b’avaira (a mitzvah that came because of a sin), like retuning a stolen object which is also a mitzvah without a brocha. There is a problem with this answer as well, because, according to Chassidus and Kabbalah, teshuvah is not necessarily done for a sin, teshuvah is returning to Hashem and elevating oneself to a higher and higher level.
Still others say the reason we don’t make a brocha on teshuvah is because real teshuvah needs to take please in the heart and we don’t make a brocha on things that are contingent on the heart. For example, we only make a brocha on the destruction of chometz, not on the nullification, which is an internal feeling, not an action.
Teshuvah is a mitzvah like all others – it needs to be done with happiness. That is why, in many shuls, the chazzan and the congregation sing the Ashamnu (confessions) out loud, to proclaim our joy in the mitzvah of teshuvah and the eternal ability of a Jew to return to Hashem.