Is there a specific posture for supplication, or is it permissible for a Muslim to supplicate in any posture? And is lying on the bed while supplicating considered ill-mannered towards Allah?

Supplication (Du'a) is a prescribed remembrance in all states. It has been authentically reported that the Prophet, peace and blessings be upon him, used to remember Allah in all circumstances. There is no harm in supplicating while on the bed. Indeed, the two Sheikhs (Bukhari and Muslim) reported that the Prophet, peace and blessings be upon him, said: "When you go to your bed, perform ablution as you do for prayer, then lie down on your right side and say: 'O Allah, I surrender my soul to You...'" to the end of the supplication. It is also mentioned in Sahih Muslim that he used to supplicate before sleeping. Allah the Almighty has praised those who remember Him standing, sitting, and on their sides.