The request is directed to the King. There are several reasons why this was appropriate. On one level, it was appropriate because they were asking their father. It is also more than appropriate, but even required, for another reason.
Note that at this time there is a separation between the political and the religious aspects of life in Zarahemla. Alma is the religious leader, but this request to perform a religious labor does not go to Alma, but rather to Mosiah. While Mosiah was a religious man, and righteous enough to be able to use the interpreters (Mosiah 8:13), he is not the head of the church. One might expect that a request of a religious nature should be directed through the head of the church.
The request comes to Mosiah as king precisely because it involves the Lamanites. The preaching to the Lamanites could have any number of consequences.
Should there be an unfavorable reception, the sons of Mosiah could be captured and held for ransom, or at least sacrificed. The preaching to the Lamanites might be seen by the Lamanites as a spying mission, and the tentative relationships between Zarahemla and the Lamanites might be thrown into disarray. In other words, the very actions proposed by the sons of Mosiah had numerous potential political ramifications, including the positive possibility of improving relationships. It is for this important reason that it was most appropriate to bring the issue to Mosiah as king.
While Mosiah was certainly the king, and was certainly more concerned with the political than the religious, that made him no less a righteous leader, as witnessed by his action. Rather than make the decision on his own, Mosiah consults the Lord, and the Lord tells him that the mission will be beneficial. This is the true role of a political leader, to counsel with the Lord for the good of his people, and to be sufficiently righteous to be worthy of receiving an answer.