Is it permissible to ask for help from someone in the army to facilitate appointment as a recruit instead of an officer, knowing that no bribe or any other compensation will be paid?

It is permissible for one who is entitled to a right to seek it through legitimate means, including interceding through a person to obtain it. This falls under the category of good intercession, which the Sharia encourages and for which the intercessor is rewarded, as Allah Almighty says: "Whoever intercedes with a good intercession will have a share from it," and as the Prophet, peace and blessings be upon him, said: "Intercede, and you will be rewarded." The intercessor is asked to do so seeking the pleasure of Allah, without taking any reward. Based on this, there is no impediment to using intercession to obtain an appointment as a soldier if that is a right.