Does any Buddhist practice consists of supplication, or is there too much merit being exchanged for that?
Supplication (also known as petitioning) is a form of prayer, wherein one party humbly or earnestly asks another party to provide something, either for the party who is doing the supplicating (e.g., "Please spare my life.") or on behalf of someone else (e.g., "Please spare my child's life.").
If there was a practice like that in Buddhism, it doesn't come from the Buddha's teachings. One practices loving-kindness by making a wish(not a prayer) such as "May I be happy" or "May the world be at peace" or "May my enemy be well" and so on. The purpose of this practice is to help negate states of anger in the practitioner thus benefiting everyone. No God, Buddha or anyone grants the wish. It all happens in our own minds. If one doesn't understand this then one might mistake practice for ritual as well. For example, one doesn't pray but pays homage to the Buddha, Dhamma and Sangha in mindful rememberance. This practice is for the benefit of the practitioner by way of building humble states of mind and when one is humble, that will benefit others as well.
All practices that come from the Buddha's teaching are like this: Benefiting both the practitioner and the beings the practitioner comes into contact with. We practice to build wholesome states for ourselves. The less we are unwholesome the less others will suffer because of us and the more they will be happy when they come into contact with us. We help others by helping ourselves. This is a good kind of selfishness :)
Perhaps the Pure Land schools of Buddhism come closest to the notion of supplication. To quote wikipedia
it is one's faith in the salvific compassion of the Buddha Amitabha, [..] and the earnest wish to enter the Buddha's happy land, that is said to bring deliverance into Buddha Amitabha's Western Paradise
However i believe that even in these schools it isn't a simple matter of just giving oneself over to another power. For instance in Shin Buddhism (also know as True Pure Land School) the Nembutsu is chanted as an expression of gratitude rather than a request for divine assistance.
On the flip side to this, even Zen practitioners bow to their cushions. So even in Zen, where the emphasis is very much on your own endeavors, recognition is regularly given to something outside of oneself. You could even say that this was almost supplication.
Supplication does appear in Buddhism, even though there is less emphasis on it than in Christian teachings: Buddhists are less likely to e.g. put their successes down to being blessed.
Just an example quote form scholarly research:
According to the general teachings [rather than Vajrayana] of Buddhism only the fully awakened one, the Buddha, can be the object of supplication, veneration and devotion.