The Third Amendment (one often forgotten) is sufficient to invalidate the use of drones.
"No Soldier shall, in time of peace be quartered in any house, without the consent of the Owner, nor in time of war, but in a manner to be prescribed by law."
When one examines the historical context surrounding the Third Amendment, they will quickly realize that it was meant to stop Police Oppression by standing armies. During the Colonial Era, Police Oppression could only be accomplished by large standing armies, which required many resources and mouths to feed. As such, these standing armies would have to lodge their soldiers within the homes of the local population, and almost always drain the local resources and food supply in order to keep themselves well nourished and prepared.
The Third Amendment today, like all other Amendments, needs to modernized for the 21st Century. A government can easily establish a Police State with just drones alone, especially if they are equipped with Non-Lethal Weaponry to suppress Freedom of Assembly. This is precisely why the Third Amendment was written.
On May 3, 1982 in the case Engblom v. Carey, the word "Soldier" was interpreted to mean "Any Agent of the State." Drones are without any doubt an Agent of the State, and furthermore, they are a Military Agent of the State. They are also being quartered over American skies without Consent.
The final clause, "but in a manner to be prescribed by law," implies that only the State Government (if not the Local Government) may Consent to the police occupation of their own State/County/Town, which may be necessary in dire circumstances.
Finally, it appears that there must be an official Declaration of War by Congress for even the State/Local Governments to waive 3rd Amendment protections. The United States has not had any official declarations of war since WWII, which means the People have never Consented to War. If they have not Consented to War, then their 3rd Amendment can not be waived, regardless of the fact that we are actually fighting several wars.