Are there any petitionary korbanot? Today we have many petitionary prayers, but I cannot think of any similar korbanot (for health, prosperity, etc.)
Kol tuv,
Yehuda
To answer your question one must look more closely at the mechanism of Tefilah and Korbanos.
The petition in Tefilah is not merely a request list; it is a way for a person to realize that all of his needs are provided by Hash-m. This brings a person closer to Hash-m through the realization that he is beholden to Him.
A Korban Nedavah is a Doron (freewill gift), and not a petition. The Ovdei Avodah Zarah offered sacrifices because they believed in appeasing the gods. L'Havdil, we know that Hash-m has no need for the sacrifice. Rather, it is an opportunity to become closer through our expression of "giving." Therefore, although by nature it is not petitionary, it relates to Tefilah in the desired result.
D. Zupnik
What about a Noder b'Eis Tzarah (see Tosfos Chulin 2b DH Aval Amar, cited in YD 203:5)?
Kol Tuv,
Yitzchok Zirkind
As Reb Yitzchok commented, Yakov asked Hashem to protect him in the house of Lavan, vowing that if Hashem protects him he will "make this place a Beis Hashem", and offer a Nesech in gratitude. That would seem to qualify as petitionary. A similar vow was taken by the Jewish Nation when they were attacked by Amalek (in the end of Parashas Chukas). They pledged that if Hashem helps them to win the war they will offer all of the booty to Hekdesh (= Korbanos), see Rashi Bamidbar 21:2. See also Nazir 10a, Nedarim 28a, where a person makes an object Hekdesh in prayer that Hashem grant his wish about that object.
I would think that the Se'ir la'Azazel is of the same category as the "Selach Lanu" prayer, in which we ask Hashem for forgiveness. The Se'ir is a way of requesting that Hashem forgive all of our sins. For that matter, every Korban Chatas is like that; we are beseeching Hashem to forgive our sins.
Mordecai Kornfeld