If you make it comparable, and ask who could send your nephews to Afghanistan right now, then again - darn near anyone in their chain of command.
For an OPLAN, the document is a TPFDD: Time-Phased Force Deployment Data. It may have a different name under contingency planning, and I don't know which is being used for Afghanistan right now. But the principle remains: the document creates slots, like job openings, if you will. And anyone with the qualifications can be plugged into one of those slots. The document is created under the SecDef's authority, IIRC. That honestly isn't something I had to deal with.
Once created, a commander can put one of their people into it, or is tasked to commit so many people to go. In some cases, and entire squadron would be committed. I had limited experience with TPFDDs, other than in some static forms.
Lots of people could sign the orders - pretty much any commander in the chain. Since Congress funds it, it is authorized by Congress via the military. But no, the President of the USA doesn't sign the orders.
And as long as Obama is accepted by the government as President, he could sign the orders if he wished. The legality of the orders would remain, because he signs them as President, not as an individual. Just as Obamacare would remain a law if Obama was found to be Kenyan-born, so would any military orders he signed (if he could be found to sign them). It is the office that has the authority, not the individual.
In this case:
